user2495: Can we consider 'tathagatagarbha' and 'dhammakaya' as synonyms? If not, what is the difference between those words.

user2265: Yes they are synonyms but like every synonym there is a fine difference: the dharmakaya (dharmmakaya) refer to the non-physical component of the Trikaya
I find the dharmakaya to be a more useful understanding because it helps to contrasts the mental component of Buddhist practice with the physical component of Buddhist/Taoist practice (chi channels, chakras, kundalini, warmth, Taoist practices, etc.). This can help your practice because one principle of Mind is that the  body and mind are in sync and follow each other: if you train/purify the body, you also train/purify the mind... to a certain point. This rupakaya development has its limits and that is why the dharmakaya is helpful to be contrasted. The rupakaya goes very deep into the physical powers and improvements listed in Buddhism from just basic health to creating full on Taoist yang shen bodies (nirmanakaya). Anyway, it is more important to understand the contrast between the rupakaya and dharmakaya and focus on cultivating dharmakaya, because this is literally Reality. Ways to cultivate this tend to be non-physical, which in terms of self-actualization, means: achieving samadhis, prajnas, and stages described in the texts (through whatever method).
That is the lesson that the word "dharmakaya" teaches, which is why I like that term to help understand different practices.
The Tathagatagarbha refers to the "womb" of reality/"Buddha"... basically Buddha-nature, basically dharmakaya.
In other sense: the Tathagatagarbha refers to a group of sutras for the most part, the Tathagatagarbha Sutras, but also to the Buddha-nature itself.
Nonetheless, it is disputed whether buddha-nature = Tathagatagarbha: the tathagatagarbha is thus described in Wikipedia:
According to Wikipedia, the Tathagatagarbha:

and the Buddha-nature have not exactly the same meaning, in the Buddhist tradition they became equated. In
  the Angulimaliya Sūtra and in the Mahāyāna Mahāparinirvāṇa Sūtra the
  terms "Buddha-nature" (Buddha-dhātu) and "tathāgatagarbha" are
  synonyms.
All are agreed that the tathāgatagarbha is an immortal, inherent
  transcendental essence or potency and that it resides in a concealed
  state (concealed by mental and behavioural negativities) in every
  single being, even the worst - the icchantika.[citation needed]
Although attempts are made in the Buddhist sutras to explain the
  tathāgatagarbha, it remains ultimately mysterious and allegedly
  unfathomable to the ordinary, unawakened person, being only fully
  knowable by perfect Buddhas themselves.
The tathāgatagarbha itself needs no cultivation, only uncovering or
  dis-covery, as it is already present and perfect within each being:
An unknown treasure exists under the home of a poor person that must
  be uncovered through removing obstructive dirt, yielding the treasure
  that always was there. Just as the treasure already exists and thus
  requires no further fashioning, so the matrix-of-one-gone-thus [i.e.
  the tathāgatagarbha], endowed with ultimate buddha qualities, already
  dwells within each sentient being and needs only to be freed from
  defilements.[8]
Charles Muller comments that the tathagatagarbha is the mind's
  original pure nature and has neither a point of origination nor a
  point of cessation: 'tathagatagarbha expresses the already perfect
  aspect of the original nature of the mind that is clear and pure
  without arising or cessation.'[9]
The tathāgatagarbha is the ultimate, pure, ungraspable, inconceivable,
  irreducible, unassailable, boundless, true and deathless quintessence
  of the Buddha's emancipatory reality, the very core of his sublime
  nature.

I agree with the above that there is a small difference:
My linguistic analysis is that "garbha" basically means womb, a word that emphasizes the physical, whereas the "tathagata" is referring more to the formless component "Buddha-nature."
The term Trikaya, which is what the dhammakaya and rupakaya are collectively called, is more equatable to the term Tathagatagarbha. Dhammakaya refers more to just the mental component of the achievements while the Tathagatagarbha refers vaguely to the Buddha-nature, the dharmakaya.. especially as described in the respective Tathagatagarbha sutras.
Thus... yes I would say they are essentially synonyms but one is more specific.

user6994: No, they are not synonyms. They are mutually exclusive: If it is 'tathagatagarbha', then it is not 'dhammakaya'. If it is 'dhammakaya', it is not 'tathagatagarbha'.
Tathagatagarbha is 'suchness with defilement', which is the emptiness of the minds of sentient beings.
Dharmakaya is 'suchness without defilement', which is the emptiness of the omniscient mind of a buddha. (more specifically any of the 'nature truth body', according to Prasangika). Although they are undifferenciated in nature (pure by nature), Dharmakaya and Tathagatagarbha are not the same.
As Gyaltsab-Je states in his commentary to Maitreya's Sublime Continuum
differentiating the Lineage of the Three Jewels:

Suchness with defilement is the basic element not liberated from the
  obscurations of mental afflictions, and is known as tathagata
  essence." By nature it is pure and is the ultimate truth. The term
  "tathagata essence" is not used anywhere by either the Tathagata
  Essence Sutra or this treatise and its commentary, which explain the
  thinking of that sutra, to teach the  twofold purity [i.e. the
  dharmakaya endowed with the twofold purity]. Therefore, to apply the
  ultimate dharmakaya to "tathagata essence" is a mere fabrication


2495: <p>Can we consider 'tathagatagarbha' and 'dhammakaya' as synonyms? If not, what is the difference between those words.</p>
 2265: <p>Yes they are synonyms but like every synonym there is a fine difference: the <a href="https://en.wikipedia.org/wiki/Dharmak%C4%81ya#Qualities" rel="nofollow">dharmakaya</a> (dharmmakaya) refer to the non-physical component of the Trikaya</p>

<p>I find the dharmakaya to be a more useful understanding because it helps to contrasts the mental component of Buddhist practice with the physical component of Buddhist/Taoist practice (chi channels, chakras, kundalini, warmth, Taoist practices, etc.). This can help your practice because one principle of Mind is that the  body and mind are in sync and follow each other: if you train/purify the body, you also train/purify the mind... to a certain point. This rupakaya development has its limits and that is why the dharmakaya is helpful to be contrasted. The rupakaya goes very deep into the physical powers and improvements listed in Buddhism from just basic health to creating full on Taoist yang shen bodies (nirmanakaya). Anyway, it is more important to understand the contrast between the rupakaya and dharmakaya and focus on cultivating dharmakaya, because this is literally Reality. <strong>Ways to cultivate this tend to be non-physical, which in terms of self-actualization, means: achieving samadhis, prajnas, and stages described in the texts (through whatever method).</strong></p>

<p>That is the lesson that the word "dharmakaya" teaches, which is why I like that term to help understand different practices.</p>

<p>The <a href="https://en.wikipedia.org/wiki/Tath%C4%81gatagarbha_s%C5%ABtras#Tathagatagarbha_and_Buddha-nature" rel="nofollow">Tathagatagarbha</a> refers to the "womb" of reality/"Buddha"... basically Buddha-nature, basically dharmakaya.
In other sense: the Tathagatagarbha refers to a group of sutras for the most part, the Tathagatagarbha Sutras, but also to the Buddha-nature itself.</p>

<p>Nonetheless, it is disputed whether buddha-nature = Tathagatagarbha: the tathagatagarbha is thus described in Wikipedia:</p>

<p>According to Wikipedia, the Tathagatagarbha:</p>

<blockquote>
  <p>and the Buddha-nature have not exactly the same meaning, in the Buddhist tradition they became equated. In
  the Angulimaliya Sūtra and in the Mahāyāna Mahāparinirvāṇa Sūtra the
  terms "Buddha-nature" (Buddha-dhātu) and "tathāgatagarbha" are
  synonyms.</p>
  
  <p>All are agreed that the tathāgatagarbha is an immortal, inherent
  transcendental essence or potency and that it resides in a concealed
  state (concealed by mental and behavioural negativities) in every
  single being, even the worst - the icchantika.[citation needed]</p>
  
  <p>Although attempts are made in the Buddhist sutras to explain the
  tathāgatagarbha, it remains ultimately mysterious and allegedly
  unfathomable to the ordinary, unawakened person, being only fully
  knowable by perfect Buddhas themselves.</p>
  
  <p>The tathāgatagarbha itself needs no cultivation, only uncovering or
  dis-covery, as it is already present and perfect within each being:</p>
  
  <p>An unknown treasure exists under the home of a poor person that must
  be uncovered through removing obstructive dirt, yielding the treasure
  that always was there. Just as the treasure already exists and thus
  requires no further fashioning, so the matrix-of-one-gone-thus [i.e.
  the tathāgatagarbha], endowed with ultimate buddha qualities, already
  dwells within each sentient being and needs only to be freed from
  defilements.[8]</p>
  
  <p>Charles Muller comments that the tathagatagarbha is the mind's
  original pure nature and has neither a point of origination nor a
  point of cessation: 'tathagatagarbha expresses the already perfect
  aspect of the original nature of the mind that is clear and pure
  without arising or cessation.'[9]</p>
  
  <p>The tathāgatagarbha is the ultimate, pure, ungraspable, inconceivable,
  irreducible, unassailable, boundless, true and deathless quintessence
  of the Buddha's emancipatory reality, the very core of his sublime
  nature.</p>
</blockquote>

<p>I agree with the above that there is a small difference:</p>

<p>My linguistic analysis is that "garbha" basically means womb, a word that emphasizes the <em>physical</em>, whereas the "tathagata" is referring more to the formless component "Buddha-nature."</p>

<p>The term Trikaya, which is what the dhammakaya and rupakaya are collectively called, is more equatable to the term Tathagatagarbha. Dhammakaya refers more to just the mental component of the achievements while the Tathagatagarbha refers vaguely to the Buddha-nature, the dharmakaya.. especially as described in the respective Tathagatagarbha sutras.</p>

<p>Thus... yes I would say they are essentially synonyms but one is more specific.</p>
 6994: <p>No, they are not synonyms. They are mutually exclusive: If it is 'tathagatagarbha', then it is not 'dhammakaya'. If it is 'dhammakaya', it is not 'tathagatagarbha'.</p>

<p>Tathagatagarbha is 'suchness with defilement', which is the emptiness of the minds of sentient beings.</p>

<p>Dharmakaya is 'suchness without defilement', which is the emptiness of the omniscient mind of a buddha. (more specifically any of the 'nature truth body', according to Prasangika). Although they are undifferenciated in nature (pure by nature), Dharmakaya and Tathagatagarbha are not the same.</p>

<p>As Gyaltsab-Je states in his commentary to Maitreya's <em>Sublime Continuum
differentiating the Lineage of the Three Jewels</em>:</p>

<blockquote>
  <p>Suchness with defilement is the basic element not liberated from the
  obscurations of mental afflictions, and is known as tathagata
  essence." By nature it is pure and is the ultimate truth. The term
  "tathagata essence" is not used anywhere by either the Tathagata
  Essence Sutra or this treatise and its commentary, which explain the
  thinking of that sutra, to teach the  twofold purity [i.e. the
  dharmakaya endowed with the twofold purity]. Therefore, to apply the
  ultimate dharmakaya to "tathagata essence" is a mere fabrication</p>
</blockquote>
