axiom vs saying

axiom and saying both are nouns.

Word	Noun	Adjective	Verb	Adverb
axiom	Yes	No	No	No
saying	Yes	No	No	No

As nouns, saying is a hypernym of axiom; that is, saying is a word with a broader meaning than axiom:

axiom: a saying that is widely accepted on its own merits
saying: a word or phrase that particular people use in particular situations

Other hypernyms of axiom include expression, locution.

axiom (noun)	saying (noun)
a saying that is widely accepted on its own merits	a word or phrase that particular people use in particular situations
(logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident