woman vs adult

woman and adult both are nouns.

Word	Noun	Adjective	Verb	Adverb
woman	Yes	No	No	No
adult	Yes	Yes	No	No

As nouns, adult is a hypernym of woman; that is, adult is a word with a broader meaning than woman:

Other hypernyms of woman include grownup, female, female person.

woman (noun)	adult (noun)
women as a class	any mature animal
an adult female person (as opposed to a man)	a fully developed person from maturity onward
a female person who plays a significant role (wife or mistress or girlfriend) in the life of a particular man

woman (adjective)	adult (adjective)
	(of animals) fully developed
	designed to arouse lust