The difference between 'maxim' and 'proverb'?

From my computer’s Dashboard dictionary:

“Once burned, twice shy” is an old saying about learning from your mistakes. In fact, sayings—a term used to describe any current or habitual expression of wisdom or truth—are a dime a dozen.

Proverbs—sayings that are well known and often repeated, usually expressing metaphorically a truth based on common sense or practical experience—are just as plentiful (her favorite proverb was “A stitch in time saves nine” ).

An adage is a time-honored and widely known proverb, such as “Where’s there’s smoke, there’s fire.”

A maxim offers a rule of conduct or action in the form of a proverb, such as “Neither a borrower nor a lender be.”

Epigram and epigraph are often confused, but their meanings are quite separate. An epigram is a terse, witty, or satirical statement that often relies on a paradox for its effect (Oscar Wilde’s well-known epigram that “The only way to get rid of temptation is to yield to it” ). An epigraph, on the other hand, is a brief quotation used to introduce a piece of writing (he used a quote from T. S. Eliot as the epigraph to his new novel).

An aphorism requires a little more thought than an epigram, since it aims to be profound rather than witty (as one of Solomon’s aphorisms warns, “Better is a living dog than a dead lion” ).

An apothegm is a pointed and often startling aphorism, such as Samuel Johnson’s remark that “Patriotism is the last refuge of a scoundrel.”