What is the difference between hyperbole and metaphor?

A hyperbole is an exaggeration of something, but a metaphor is comparing something (not necessarily an exaggeration).

Hyperbole example:

She's as skinny as a toothpick.

Obviously, the girl is not as skinny as a toothpick, however, it is being used to show that she is extremely thin.

Metaphor example:

She is a big baby.

Here you are comparing her to a baby, not because she is one, but because she may act like on at times.

A hyperbole is an exaggeration not to be taken literally for emphasis or effect.

That sweater shrunk 10 sizes at least. (OK, the sweater probably shrunk, but I doubt it shrunk that many.)

A metaphor is a comparison between two unrelated objects. For instance, beating a dead horse to mean that something has been said or done too many times.