modern dance vs stage dancing

modern dance and stage dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
modern dance	Yes	No	No	No
stage dancing	Yes	No	No	No

As nouns, stage dancing is a hypernym of modern dance; that is, stage dancing is a word with a broader meaning than modern dance:

modern dance: a style of theatrical dancing that is not as restricted as classical ballet; movements are expressive of feelings
stage dancing: a show involving artistic dancing

Other hypernyms of modern dance include choreography.

modern dance (noun)	stage dancing (noun)
a style of theatrical dancing that is not as restricted as classical ballet; movements are expressive of feelings	a show involving artistic dancing