An idiom is a phrase that has a meaning of its own that cannot be understood from the meanings of its individual words.

Here are some examples of idioms:

to be fed up with means to be tired and annoyed with something that has been happening for too long

to rub someone the wrong way means to irritate someone

by the skin of your teeth means that something was successful, but only just barely. “She passed the test by the skin of her teeth” means she almost didn’t pass.

A proverb is a short popular saying that gives advice about how people should behave or that expresses a belief that is generally thought to be true.

Here are some examples:

Don’t cry over spilled milk.

Those who live in glass houses shouldn’t throw stones.

A stitch in time saves nine.

Like idioms, proverbs often have a meaning that is greater than the meaning of the individual words put together, but in a different way than idioms.

The literal meaning of an idiom usually doesn’t make sense, and idioms can be almost impossible to understand unless you have learned or heard them before.

The literal meaning of a proverb such as “Don’t cry over spilled milk” does makes sense on its own, but it’s not until you apply this meaning to a broader set of situations that you understand the real point of the proverb.

For example, “Don’t cry over spilled milk” means “Don’t get upset over something that has already been done.

It’s too late to worry about it now, just get on with your life.

References:

https://www.youtube.com/watch?v=EpRjtiisJ5M

http://www.learnersdictionary.com/qa/what-s-the-difference-between-idioms-and-proverbs