dancer vs performing artist

dancer and performing artist both are nouns.

Word	Noun	Adjective	Verb	Adverb
dancer	Yes	No	No	No
performing artist	Yes	No	No	No

As nouns, performing artist is a hypernym of dancer; that is, performing artist is a word with a broader meaning than dancer:

dancer: a performer who dances professionally
performing artist: an entertainer who performs a dramatic or musical work for an audience

Other hypernyms of dancer include performer.

dancer (noun)	performing artist (noun)
a performer who dances professionally	an entertainer who performs a dramatic or musical work for an audience
a person who participates in a social gathering arranged for dancing (as a ball)