slang word vs saying

slang word and saying both are nouns.

Word	Noun	Adjective	Verb	Adverb
slang word	Yes	No	No	No
saying	Yes	No	No	No

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

slang word: a word or expression that is outside of standard usage and is very informal
saying: a word or phrase that particular people use in particular situations

Other hypernyms of slang word include expression, locution.

slang word (noun)	saying (noun)
a word or expression that is outside of standard usage and is very informal	a word or phrase that particular people use in particular situations