wig vs hairpiece

wig and hairpiece both are nouns.

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

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

wig: hairpiece covering the head and made of real or synthetic hair
hairpiece: a covering or bunch of human or artificial hair used for disguise or adornment

Other hypernyms of wig include false hair, postiche.

wig (noun)	hairpiece (noun)
hairpiece covering the head and made of real or synthetic hair	a covering or bunch of human or artificial hair used for disguise or adornment
British slang for a scolding