hairpiece vs false hair

hairpiece and false hair both are nouns.

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

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

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

Other synonyms of hairpiece include postiche.

hairpiece (noun)	false hair (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