first lady vs wife

first lady and wife both are nouns.

Word	Noun	Adjective	Verb	Adverb
first lady	Yes	No	No	No
wife	Yes	No	No	No

As nouns, wife is a hypernym of first lady; that is, wife is a word with a broader meaning than first lady:

Other hypernyms of first lady include married woman.

first lady (noun)	wife (noun)
the wife of a chief executive	a married woman; a man's partner in marriage
the leading woman in an art or profession