round dancing vs ballroom dance

round dancing and ballroom dance both are nouns.

Word	Noun	Adjective	Verb	Adverb
round dancing	Yes	No	No	No
ballroom dance	Yes	No	No	No

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

round dancing: a ballroom dance characterized by revolving movement
ballroom dance: any of a variety of social dances performed by couples in a ballroom

Other hypernyms of round dancing include ballroom dancing.

round dancing (noun)	ballroom dance (noun)
a ballroom dance characterized by revolving movement	any of a variety of social dances performed by couples in a ballroom