cotillion vs ballroom dancing

cotillion and ballroom dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
cotillion	Yes	No	No	No
ballroom dancing	Yes	No	No	No

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

cotillion: a lively dance originating in France in the 18th century
ballroom dancing: any of a variety of social dances performed by couples in a ballroom

Other hypernyms of cotillion include ballroom dance.

cotillion (noun)	ballroom dancing (noun)
a lively dance originating in France in the 18th century	any of a variety of social dances performed by couples in a ballroom
a ball at which young ladies are presented to society