madwoman vs adult female

madwoman and adult female both are nouns.

Word	Noun	Adjective	Verb	Adverb
madwoman	Yes	No	No	No
adult female	Yes	No	No	No

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

Other hypernyms of madwoman include lunatic, maniac, woman.

madwoman (noun)	adult female (noun)
a woman lunatic	an adult female person (as opposed to a man)