toupee vs hairpiece

toupee and hairpiece both are nouns.

Word	Noun	Adjective	Verb	Adverb
toupee	Yes	No	No	No
hairpiece	Yes	No	No	No

As nouns, hairpiece is a hypernym of toupee; that is, hairpiece is a word with a broader meaning than toupee:

toupee: a small hairpiece to cover partial baldness
hairpiece: a covering or bunch of human or artificial hair used for disguise or adornment

Other hypernyms of toupee include false hair, postiche.

toupee (noun)	hairpiece (noun)
a small hairpiece to cover partial baldness	a covering or bunch of human or artificial hair used for disguise or adornment