I'm sure you've checked the dictionary, so you'll forgive me if I don't.

An idiom is a fixed expression whose meaning differs from what you'd come up with by taking the standard meanings of the individual words.  (Don't ask me for the difference between a "fixed expression" and an "idiom.")

A proverb is a wise saying, which does not necessarily include an idiom, but may.  "An ounce of prevention is worth a pound of cure."  "A stitch in time saves nine."  "Waste not, want not."  "A watched kettle never boils."  These are sometimes a bit clever, but a clever person can figure out what they mean.

Well, I spaced out trying to think of examples of idioms, so I borrowed this one from Wickipedia:  "to kick the bucket"  =  "to die."  You could never figure that out by looking up the words in the dictionary.

On the way there, I came across huge lists of common phrases and expressions which were claimed to be "idioms."  They do not fit the definition I gave above.  I'd just call them common phrases and expressions.  Many were proverbs.  (Others may disagree.)

Naturally, this made me break my word and visit my dictionary.  It generally supports the definition I gave above.

I often think of "idioms" as fixed expressions in which the words appear to break some rule of grammar.  That is, the useage is an idiom, but does not necessarily convey a complete thought.  "Kick the bucket" would not be of this type.

The great bulk of web sites seem to define "idioms" as "common sayings."  I guess it would be foolish for me to advise you otherwise.