woman's doctor vs specialist

woman's doctor and specialist both are nouns.

Word	Noun	Adjective	Verb	Adverb
woman's doctor	Yes	No	No	No
specialist	Yes	No	No	No

As nouns, specialist is a hypernym of woman's doctor; that is, specialist is a word with a broader meaning than woman's doctor:

Other hypernyms of woman's doctor include medical specialist.

woman's doctor (noun)	specialist (noun)
a specialist in gynecology	an expert who is devoted to one occupation or branch of learning
	practices one branch of medicine