What's the difference between idioms and proverbs?

Idioms are a group of words that wouldn't make sense if they were taken literally, and are given a different meaning when put together. A proverb is basically just a saying.

An idiom is an expression with a meaning or grammatical structure that can't be understood based on its individual words. It seems like a wrong usage if you think about it, but is widely accepted anyway. It is typically just a few words, not as much as a whole sentence. "Head over heels" meaning upside-down is an idiom.

A proverb is a brief piece of "folk wisdom". Typically one complete sentence, it expresses some form of advice; many people can quote it exactly, but typically it is not known where it came from originally. (The Bible includes a book called Proverbs, and I presume some of them come from there.
Example: "Marry at haste, repent at leisure."

An idiom is a natural manner of speaking within a culture and time period. A proverb is some wisdom accrued from the past and passed on down the generations to give advice.