user8157: DN 2 states:

When this was said, Ajita Kesakambalin said to me, 'Great king, there is nothing given, nothing offered, nothing sacrificed. There is no
  fruit or result of good or bad actions. There is no this world, no
  other world, no mother, no father, no spontaneously reborn beings; no
  brahmans or contemplatives who, faring rightly and practicing rightly,
  proclaim this world and the other after having directly known and
  realized it for themselves. A person is a composite of four primary
  elements. At death, the earth (in the body) returns to and merges with
  the (external) earth-substance. The fire returns to and merges with
  the external fire-substance. The liquid returns to and merges with the
  external liquid-substance. The wind returns to and merges with the
  external wind-substance. The sense-faculties scatter into space. Four
  men, with the bier as the fifth, carry the corpse. Its eulogies are
  sounded only as far as the charnel ground. The bones turn
  pigeon-colored. The offerings end in ashes. Generosity is taught by
  idiots. The words of those who speak of existence after death are
  false, empty chatter. With the break-up of the body, the wise and the
  foolish alike are annihilated, destroyed. They do not exist after
  death.'

Ajita Kesakambali is similar to another wrong view, found in DN 1, namely:

Herein, bhikkhus, a certain recluse or a brahmin asserts the following doctrine and view: ‘The self, good sir, has material form;
  it is composed of the four primary elements and originates from father
  and mother. Since this self, good sir, is annihilated and destroyed
  with the breakup of the body and does not exist after death, at this
  point the self is completely annihilated.’ In this way some proclaim
  the annihilation, destruction, and extermination of an existent
  being.

Now the Buddha taught extensively about the elements, such as in MN 115 & MN 140. 
Or in MN 43, Sariputta mentions how the sense faculties are scattered at the termination of life, as follows:

Yvāyaṃ, āvuso, mato kālaṅkato tassa kāyasaṅkhārā niruddhā paṭippassaddhā, vacīsaṅkhārā niruddhāniruddha paṭippassaddhā, cittasaṅkhārā niruddhā
  paṭippassaddhā, āyu parikkhīṇo, usmā vūpasantā, indriyāni
  paribhinnāni.
In the case of the one who is dead, who has completed his time, his bodily fabrications have ceased & subsided, his verbal fabrications
  ... his mental fabrications have ceased & subsided, his vitality is
  exhausted, his heat subsided, & his faculties are scattered.

How do the teachings of Ajita Kesakambali & other annihilationists compare to those of the Buddha? What makes them different? 

user10100: Perfect right view of buddha is:
There are fruits, and effects of good or bad actions, causes (paṭiccasamuppada and paṭiccasamuppanna). Elements such as 89 consciousnesses, 52 mind factors, and 28 matters, are causes and effects of each others. And one effect is arised by causes, so if when causes will change to be impossible to make that effect arise, that effect follow vanish or impossible to be, to arise, too (anicca dukkha anatta).
But the other teachers have not perfect right view like the buddha. Such as ajitakesakambala's view that his elements, causes,  have not their effects, fruits and results.
I am sorry. I write this answer on mobile phone. So it maybe not too perfect.

user471: The Buddha's teachings are different from what Ajita Kesakambali taught.
The Buddha realized this (from MN 19):

"When the mind was thus concentrated, purified, bright, unblemished,
  rid of defilement, pliant, malleable, steady, & attained to
  imperturbability, I directed it to the knowledge of recollecting my
  past lives. I recollected my manifold past lives, i.e., one birth,
  two... five, ten... fifty, a hundred, a thousand, a hundred thousand,
  many eons of cosmic contraction, many eons of cosmic expansion, many
  eons of cosmic contraction & expansion: 'There I had such a name,
  belonged to such a clan, had such an appearance. Such was my food,
  such my experience of pleasure & pain, such the end of my life.
  Passing away from that state, I re-arose there. There too I had such a
  name, belonged to such a clan, had such an appearance. Such was my
  food, such my experience of pleasure & pain, such the end of my life.
  Passing away from that state, I re-arose here.' Thus I remembered my
  manifold past lives in their modes & details. .....
"When the mind was thus concentrated, purified, bright, unblemished,
  rid of defilement, pliant, malleable, steady, & attained to
  imperturbability, I directed it to the knowledge of the passing away &
  reappearance of beings. I saw — by means of the divine eye, purified &
  surpassing the human — beings passing away & re-appearing, and I
  discerned how they are inferior & superior, beautiful & ugly,
  fortunate & unfortunate in accordance with their kamma: 'These beings
  — who were endowed with bad conduct of body, speech & mind, who
  reviled the Noble Ones, held wrong views and undertook actions under
  the influence of wrong views — with the break-up of the body, after
  death, have re-appeared in the plane of deprivation, the bad
  destination, the lower realms, in hell. But these beings — who were
  endowed with good conduct of body, speech, & mind, who did not revile
  the Noble Ones, who held right views and undertook actions under the
  influence of right views — with the break-up of the body, after death,
  have re-appeared in the good destinations, in the heavenly world.'
  Thus — by means of the divine eye, purified & surpassing the human — I
  saw beings passing away & re-appearing, and I discerned how they are
  inferior & superior, beautiful & ugly, fortunate & unfortunate in
  accordance with their kamma.

The Buddha taught his followers to reflect in this way (from AN 5.57):

“This noble disciple reflects thus: ‘I am not the only one who is the
  owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s
  origin, kamma as one’s relative, kamma as one’s resort; who will be
  the heir of whatever kamma, good or bad, that one does. All beings
  that come and go, that pass away and undergo rebirth, are owners of
  their kamma, heirs of their kamma; all have kamma as their origin,
  kamma as their relative, kamma as their resort; all will be heirs of
  whatever kamma, good or bad, that they do.’ As he often reflects on
  this theme, the path is generated. He pursues this path, develops it,
  and cultivates it. As he does so, the fetters are entirely abandoned
  and the underlying tendencies are uprooted.

From SN 12.19:

“Bhikkhus, for the fool, hindered by ignorance and fettered by
  craving, this body has originated. For the fool that ignorance has not
  been abandoned and that craving has not been utterly destroyed. For
  what reason? Because the fool has not lived the holy life for the
  complete destruction of suffering. Therefore, with the breakup of the
  body, the fool fares on to another body. Faring on to another body, he
  is not freed from birth, aging, and death; not freed from sorrow,
  lamentation, pain, displeasure, and despair; not freed from suffering,
  I say.

From SN 44.9:

“And, Master Gotama, when a being has laid down this body but has not
  yet been reborn in another body, what does Master Gotama declare to be
  its fuel on that occasion?”
“When, Vaccha, a being has laid down this body but has not yet been
  reborn in another body, I declare that it is fuelled by craving. For
  on that occasion craving is its fuel.”

Also, please note that the term "kāya" has been used in the sense of physical body for example in SN 22.56 (although I know that it can be used to mean group or collection, when combined with other things):

eye, ear, nose, tongue, body, and mind consciousness.
cakkhuviññāṇaṃ, sotaviññāṇaṃ,ghānaviññāṇaṃ, jivhāviññāṇaṃ, kāyaviññāṇaṃ, manoviññāṇaṃ

Pali words for body apart from "kāya" has been used in this context, for example "sarira" in Dhammapada 400:

Akkodhanam vatavantam
  silavantam anussadam
  dantam antimasariram1
  tamaham brumi brahmanam
Verse 400: Him I call a brahmana, who is free from anger, who
  practises austerity, who is virtuous and free from craving, who is
  controlled in his senses and for whom this body (i.e., existence) is
  the very last.
Footnote 1. antimasariram: lit., one who has the last body. This is
  his last body because he will not be reborn; he is an arahat.

From SN 15.3:

"This is the greater: the tears you have shed while transmigrating &
  wandering this long, long time — crying & weeping from being joined
  with what is displeasing, being separated from what is pleasing — not
  the water in the four great oceans.
"Long have you (repeatedly) experienced the death of a mother. The
  tears you have shed over the death of a mother while transmigrating &
  wandering this long, long time — crying & weeping from being joined
  with what is displeasing, being separated from what is pleasing — are
  greater than the water in the four great oceans.
"Long have you (repeatedly) experienced the death of a father... the
  death of a brother... the death of a sister... the death of a son...
  the death of a daughter... loss with regard to relatives... loss with
  regard to wealth... loss with regard to disease. The tears you have
  shed over loss with regard to disease while transmigrating & wandering
  this long, long time — crying & weeping from being joined with what is
  displeasing, being separated from what is pleasing — are greater than
  the water in the four great oceans.
"Why is that? From an inconstruable beginning comes transmigration. A
  beginning point is not evident, though beings hindered by ignorance
  and fettered by craving are transmigrating & wandering on. Long have
  you thus experienced stress, experienced pain, experienced loss,
  swelling the cemeteries — enough to become disenchanted with all
  fabricated things, enough to become dispassionate, enough to be
  released."

Also, please note that since some people do not accept the Digha Nikaya as genuine teachings of the Buddha (please see this question), I have quoted from all the other nikayas instead. Dhammapada is part of KN.

user11762: That quote from Ajita Kesakambalin is very similar to what Buddha taught, especially if we view his "no this, no that" as pointing that those things are selfless, devoid of substance.
The difference is that Buddha looked in wider context, not limited by the scope of the ordinary observation. For example, if something has ended as material thing, it doesn't mean its influence ended; likewise, personal human life doesn't end in borders of the living body neither spatially nor temporally. Life of the Buddha and its influence until today all over the world is an example of that. Therefore, limits of seeing "one life" are illusory, and so Buddha spoke of other lives etc. - beyond the illusory limits of ordinary consciousness.

user111: 
his mental fabrications have ceased & subsided

This refers to the mental activity in this body. It doesn't mean that there's no next life after death, unless we are talking about an Arahanth. This is why having a good teacher is important. If someone with materialistic views try to interpret this sutta by himself, he might conclude that it all ends after death.

8157: <p>DN 2 states:</p>

<blockquote>
  <p><em>When this was said, Ajita Kesakambalin said to me, 'Great king, there is nothing given, nothing offered, nothing sacrificed. There is no
  fruit or result of good or bad actions. There is no this world, no
  other world, no mother, no father, no spontaneously reborn beings; no
  brahmans or contemplatives who, faring rightly and practicing rightly,
  proclaim this world and the other after having directly known and
  realized it for themselves. A person is a composite of four primary
  elements. At death, the earth (in the body) returns to and merges with
  the (external) earth-substance. The fire returns to and merges with
  the external fire-substance. The liquid returns to and merges with the
  external liquid-substance. The wind returns to and merges with the
  external wind-substance. The sense-faculties scatter into space. Four
  men, with the bier as the fifth, carry the corpse. Its eulogies are
  sounded only as far as the charnel ground. The bones turn
  pigeon-colored. The offerings end in ashes. Generosity is taught by
  idiots. The words of those who speak of existence after death are
  false, empty chatter. With the break-up of the body, the wise and the
  foolish alike are annihilated, destroyed. They do not exist after
  death.'</em></p>
</blockquote>

<p>Ajita Kesakambali is similar to another wrong view, found in DN 1, namely:</p>

<blockquote>
  <p><em>Herein, bhikkhus, a certain recluse or a brahmin asserts the following doctrine and view: ‘The self, good sir, has material form;
  it is composed of the four primary elements and originates from father
  and mother. Since this self, good sir, is annihilated and destroyed
  with the breakup of the body and does not exist after death, at this
  point the self is completely annihilated.’ In this way some proclaim
  the annihilation, destruction, and extermination of an existent
  being</em>.</p>
</blockquote>

<p>Now the Buddha taught extensively about the elements, such as in <a href="http://www.yellowrobe.com/component/content/article/120-majjhima-nikaya/321-bahudhtuka-sutta-the-many-kinds-of-elements.html" rel="nofollow noreferrer">MN 115</a> &amp; <a href="http://www.accesstoinsight.org/tipitaka/mn/mn.140.than.html" rel="nofollow noreferrer">MN 140</a>. </p>

<p>Or in <a href="http://www.accesstoinsight.org/tipitaka/mn/mn.043.than.html" rel="nofollow noreferrer">MN 43</a>, Sariputta mentions how the sense faculties are scattered at the termination of life, as follows:</p>

<blockquote>
  <p><em>Yvāyaṃ, āvuso, mato kālaṅkato tassa kāyasaṅkhārā niruddhā paṭippassaddhā, vacīsaṅkhārā niruddhāniruddha paṭippassaddhā, cittasaṅkhārā niruddhā
  paṭippassaddhā, āyu parikkhīṇo, usmā vūpasantā, indriyāni
  paribhinnāni.</em></p>
  
  <p><em>In the case of the one who is dead, who has completed his time, his bodily fabrications have ceased &amp; subsided, his verbal fabrications
  ... his mental fabrications have ceased &amp; subsided, his vitality is
  exhausted, his heat subsided, &amp; his faculties are scattered.</em></p>
</blockquote>

<p>How do the teachings of Ajita Kesakambali &amp; other annihilationists compare to those of the Buddha? What makes them different? </p>
 10100: <p>Perfect right view of buddha is:</p>

<p>There are fruits, and effects of good or bad actions, causes (paṭiccasamuppada and paṭiccasamuppanna). Elements such as 89 consciousnesses, 52 mind factors, and 28 matters, are causes and effects of each others. And one effect is arised by causes, so if when causes will change to be impossible to make that effect arise, that effect follow vanish or impossible to be, to arise, too (anicca dukkha anatta).</p>

<p>But the other teachers have not perfect right view like the buddha. Such as ajitakesakambala's view that his elements, causes,  have not their effects, fruits and results.</p>

<p>I am sorry. I write this answer on mobile phone. So it maybe not too perfect.</p>
 471: <p>The Buddha's teachings are different from what Ajita Kesakambali taught.</p>

<p>The Buddha realized this (from <a href="https://www.accesstoinsight.org/tipitaka/mn/mn.019.than.html" rel="nofollow noreferrer">MN 19</a>):</p>

<blockquote>
  <p>"When the mind was thus concentrated, purified, bright, unblemished,
  rid of defilement, pliant, malleable, steady, &amp; attained to
  imperturbability, I directed it to the knowledge of recollecting my
  past lives. I recollected my manifold past lives, i.e., one birth,
  two... five, ten... fifty, a hundred, a thousand, a hundred thousand,
  many eons of cosmic contraction, many eons of cosmic expansion, many
  eons of cosmic contraction &amp; expansion: 'There I had <strong>such a name</strong>,
  <strong>belonged to such a clan</strong>, had <strong>such an appearance</strong>. Such was my food,
  such my experience of pleasure &amp; pain, <strong>such the end of my life</strong>.
  <strong>Passing away from that state, I re-arose there</strong>. There too I had such a
  name, belonged to such a clan, had such an appearance. Such was my
  food, such my experience of pleasure &amp; pain, such the end of my life.
  Passing away from that state, I re-arose here.' Thus I remembered my
  manifold past lives in their modes &amp; details. .....</p>
  
  <p>"When the mind was thus concentrated, purified, bright, unblemished,
  rid of defilement, pliant, malleable, steady, &amp; attained to
  imperturbability, I directed it to the knowledge of the passing away &amp;
  reappearance of beings. I saw — by means of the divine eye, purified &amp;
  surpassing the human — beings passing away &amp; re-appearing, and I
  discerned how they are inferior &amp; superior, beautiful &amp; ugly,
  fortunate &amp; unfortunate in accordance with their kamma: 'These beings
  — who were endowed with bad conduct of body, speech &amp; mind, who
  reviled the Noble Ones, held wrong views and undertook actions under
  the influence of wrong views — <strong>with the break-up of the body, after
  death, have re-appeared</strong> in the plane of deprivation, the bad
  destination, the lower realms, in hell. But these beings — who were
  endowed with good conduct of body, speech, &amp; mind, who did not revile
  the Noble Ones, who held right views and undertook actions under the
  influence of right views — <strong>with the break-up of the body, after death,
  have re-appeared</strong> in the good destinations, in the heavenly world.'
  Thus — by means of the divine eye, purified &amp; surpassing the human — <strong>I
  saw beings passing away &amp; re-appearing, and I discerned how they are
  inferior &amp; superior, beautiful &amp; ugly, fortunate &amp; unfortunate in
  accordance with their kamma.</strong></p>
</blockquote>

<p>The Buddha taught his followers to reflect in this way (from <a href="https://suttacentral.net/an5.57/en/bodhi" rel="nofollow noreferrer">AN 5.57</a>):</p>

<blockquote>
  <p>“This noble disciple reflects thus: ‘I am not the only one who is the
  owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s
  origin, kamma as one’s relative, kamma as one’s resort; who will be
  the heir of whatever kamma, good or bad, that one does. All beings
  that come and go, that pass away and undergo rebirth, are owners of
  their kamma, heirs of their kamma; all have kamma as their origin,
  kamma as their relative, kamma as their resort; all will be heirs of
  whatever kamma, good or bad, that they do.’ As he often reflects on
  this theme, the path is generated. He pursues this path, develops it,
  and cultivates it. As he does so, the fetters are entirely abandoned
  and the underlying tendencies are uprooted.</p>
</blockquote>

<p>From <a href="https://suttacentral.net/sn12.19/en/bodhi" rel="nofollow noreferrer">SN 12.19</a>:</p>

<blockquote>
  <p>“Bhikkhus, for the fool, hindered by ignorance and fettered by
  craving, this body has originated. For the fool that ignorance has not
  been abandoned and that craving has not been utterly destroyed. For
  what reason? Because the fool has not lived the holy life for the
  complete destruction of suffering. <strong>Therefore, with the breakup of the
  body, the fool fares on to another body</strong>. Faring on to another body, he
  is not freed from birth, aging, and death; not freed from sorrow,
  lamentation, pain, displeasure, and despair; not freed from suffering,
  I say.</p>
</blockquote>

<p>From <a href="https://suttacentral.net/sn44.9/en/bodhi" rel="nofollow noreferrer">SN 44.9</a>:</p>

<blockquote>
  <p>“And, Master Gotama, when a being has laid down this body but has not
  yet been reborn in another body, what does Master Gotama declare to be
  its fuel on that occasion?”</p>
  
  <p>“When, Vaccha, a being has laid down this body but has not yet been
  reborn in another body, I declare that it is fuelled by craving. For
  on that occasion craving is its fuel.”</p>
</blockquote>

<p>Also, please note that the term "kāya" has been used in the sense of physical body for example in <a href="https://suttacentral.net/sn22.56/en/sujato" rel="nofollow noreferrer">SN 22.56</a> (although I know that it can be used to mean group or collection, when combined with other things):</p>

<blockquote>
  <p>eye, ear, nose, tongue, body, and mind consciousness.<br>
  <em>cakkhuviññāṇaṃ, sotaviññāṇaṃ,ghānaviññāṇaṃ, jivhāviññāṇaṃ, <strong>kāyaviññāṇaṃ</strong>, manoviññāṇaṃ</em></p>
</blockquote>

<p>Pali words for body apart from "kāya" has been used in this context, for example "sarira" in <a href="http://www.tipitaka.net/tipitaka/dhp/verseload.php?verse=400" rel="nofollow noreferrer">Dhammapada 400</a>:</p>

<blockquote>
  <p><em>Akkodhanam vatavantam<br>
  silavantam anussadam<br>
  dantam antimasariram<sup>1</sup><br>
  tamaham brumi brahmanam</em></p>
  
  <p>Verse 400: Him I call a brahmana, who is free from anger, who
  practises austerity, who is virtuous and free from craving, who is
  controlled in his senses and for whom this body (i.e., existence) is
  the very last.</p>
  
  <p>Footnote 1. antimasariram: lit., one who has the last body. This is
  his last body because he will not be reborn; he is an arahat.</p>
</blockquote>

<p>From <a href="https://www.accesstoinsight.org/tipitaka/sn/sn15/sn15.003.than.html" rel="nofollow noreferrer">SN 15.3</a>:</p>

<blockquote>
  <p>"This is the greater: the tears you have shed while transmigrating &amp;
  wandering this long, long time — crying &amp; weeping from being joined
  with what is displeasing, being separated from what is pleasing — not
  the water in the four great oceans.</p>
  
  <p>"Long have you (repeatedly) experienced the death of a mother. The
  tears you have shed over the death of a mother while transmigrating &amp;
  wandering this long, long time — crying &amp; weeping from being joined
  with what is displeasing, being separated from what is pleasing — are
  greater than the water in the four great oceans.</p>
  
  <p>"Long have you (repeatedly) experienced the death of a father... the
  death of a brother... the death of a sister... the death of a son...
  the death of a daughter... loss with regard to relatives... loss with
  regard to wealth... loss with regard to disease. The tears you have
  shed over loss with regard to disease while transmigrating &amp; wandering
  this long, long time — crying &amp; weeping from being joined with what is
  displeasing, being separated from what is pleasing — are greater than
  the water in the four great oceans.</p>
  
  <p>"Why is that? From an inconstruable beginning comes transmigration. A
  beginning point is not evident, though beings hindered by ignorance
  and fettered by craving are transmigrating &amp; wandering on. Long have
  you thus experienced stress, experienced pain, experienced loss,
  swelling the cemeteries — enough to become disenchanted with all
  fabricated things, enough to become dispassionate, enough to be
  released."</p>
</blockquote>

<p>Also, please note that since some people do not accept the Digha Nikaya as genuine teachings of the Buddha (please see <a href="https://buddhism.stackexchange.com/q/31099/471">this question</a>), I have quoted from all the other nikayas instead. Dhammapada is part of KN.</p>
 11762: <p>That quote from Ajita Kesakambalin is very similar to what Buddha taught, especially if we view his "no this, no that" as pointing that those things are selfless, devoid of substance.</p>

<p>The difference is that Buddha looked in wider context, not limited by the scope of the ordinary observation. For example, if something has ended as material thing, it doesn't mean its influence ended; likewise, personal human life doesn't end in borders of the living body neither spatially nor temporally. Life of the Buddha and its influence until today all over the world is an example of that. Therefore, limits of seeing "one life" are illusory, and so Buddha spoke of other lives etc. - beyond the illusory limits of ordinary consciousness.</p>
 111: <blockquote>
  <p>his mental fabrications have ceased &amp; subsided</p>
</blockquote>

<p>This refers to the mental activity in this body. It doesn't mean that there's no next life after death, unless we are talking about an Arahanth. This is why having a good teacher is important. If someone with materialistic views try to interpret this sutta by himself, he might conclude that it all ends after death.</p>
