What is difference between an adage axiom aphorism epigram proverb maxim and idiom?

Axiom is a fundamental simple truth like a circle is round.

An aphorism is a short, pithy, tersely phrased statement that cleverly denotes an observation.

An epigram, should strictly just mean an inscription but since the Greeks had very clever inscriptions, it can be used more widely to pertain to any clever witty poetical statement.

A maxim is a person's .......

As you can see, they can very much be used interchangeably since they are all so similar. There's no reason to be pedantic because a Word is only defined by the way it's used. If you do want a good explanation of each, type "saying" into wikipedia. You'll find a succinct definition of each there.