What is the difference between independent and mutually exclusive events? Two events are mutually exclusive if they can't both happen.

Independent events are events where knowledge of the probability of one doesn't change the probability of the other.

Are these definitions correct? If possible, please give more than one example and counterexample.
Yes, that's fine.

Events are mutually exclusive if the occurrence of one event excludes the occurrence of the other(s).  Mutually exclusive events cannot happen at the same time. For example: when tossing a coin, the result can either be heads or tails but cannot be both.

$$\left.\begin{align}P(A\cap B) &amp;= 0 \\ P(A\cup B) &amp;= P(A)+P(B)\\ P(A\mid B)&amp;=0 \\ P(A\mid \neg B) &amp;= \frac{P(A)}{1-P(B)}\end{align}\right\}\text{ mutually exclusive }A,B$$

Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other(s).  For example: when tossing two coins, the result of one flip does not affect the result of the other.

$$\left.\begin{align}P(A\cap B) &amp;= P(A)P(B) \\ P(A\cup B) &amp;= P(A)+P(B)-P(A)P(B)\\ P(A\mid B)&amp;=P(A) \\ P(A\mid \neg B) &amp;= P(A)\end{align}\right\}\text{ independent }A,B$$

This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive.  (Events of measure zero excepted.)
Events are Independent when happening of one does not influence happening of other. Eruption of volcano on Earth and orbit of Mars do not influence each other, so are independent events.

Growth of human population and preservation of many other species are mutually exclusive, as the one can only happen if the other does not happen.

Strictly speaking, mutually exclusive does not imply that one of them must happen. If there is a large asteroid impact on Earth, then neither human population grows nor endangered species are preserved.