false hair vs wig

false hair and wig both are nouns.

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

As nouns, wig is a hyponym of false hair; that is, wig is a word with a more specific, narrower meaning than false hair:

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

Other hyponyms of false hair include Afro-wig, switch, toupe, toupee.

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