An adage is a traditional saying or proverb that states a piece of wisdom or a general truth. An adage is usually something that has been repeated for so long that it is a cliche, such as 'less is more'.
An axiom is a statement or principle that is accepted as being true, self-evidently true, or proven true by virtue of experience. An axiom can be the foundation for further study or knowledge. Axiom and adage are interchangeable and are usually sayings that have been repeated, often over many generations. Mathematicians make heavy use of axioms and maxims. An Aphorism can be a short Axiom.
An epigram is a short, witty, saying or remark expressed with brevity and sometimes paradoxically. An epigram is usually an original saying or remark, such as 'I can resist everything but temptation'. Many famous quotes are epigrams.
An idiom is a word or phrase whose meaning cannot be understood outside its cultural context. These expressions are usually figurative, such as 'cut to the chase', 'rule of thumb', etc. Idioms generally convey a casual tone and are known culturally, so other cultures may not understand the phrase or meaning. Many adages, axioms, and epigrams can also be considered idioms if not universally known.