dancing vs performing arts

dancing and performing arts both are nouns.

Word	Noun	Adjective	Verb	Adverb
dancing	Yes	No	No	No
performing arts	Yes	No	No	No

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

dancing: taking a series of rhythmical steps (and movements) in time to music
performing arts: arts or skills that require public performance

Other hypernyms of dancing include diversion, recreation.

dancing (noun)	performing arts (noun)
taking a series of rhythmical steps (and movements) in time to music	arts or skills that require public performance