wig vs false hair

wig and false hair both are nouns.

Word	Noun	Adjective	Verb	Adverb
wig	Yes	No	No	No
false hair	Yes	No	No	No

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

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

Other hypernyms of wig include hairpiece, postiche.

wig (noun)	false hair (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