false hair vs toupee

false hair and toupee both are nouns.

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

As nouns, toupee is a hyponym of false hair; that is, toupee 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
toupee: a small hairpiece to cover partial baldness

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

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