false hair vs hairpiece

false hair and hairpiece both are nouns.

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

As nouns, false hair and hairpiece are synonyms defined as:

false hair and hairpiece: a covering or bunch of human or artificial hair used for disguise or adornment

Other synonyms of false hair include postiche.

false hair (noun)	hairpiece (noun)
a covering or bunch of human or artificial hair used for disguise or adornment	a covering or bunch of human or artificial hair used for disguise or adornment