woman's hat vs pillbox

woman's hat and pillbox both are nouns.

Word	Noun	Adjective	Verb	Adverb
woman's hat	Yes	No	No	No
pillbox	Yes	No	No	No

As nouns, pillbox is a hyponym of woman's hat; that is, pillbox is a word with a more specific, narrower meaning than woman's hat:

Other hyponyms of woman's hat include church hat, cloche, picture hat, toque, turban.

woman's hat (noun)	pillbox (noun)
hats for women; the wares sold by a milliner	a small enclosed gun emplacement (usually of fortified concrete)
	a small case for holding pills
	a small round woman's hat