cotillion vs ballroom dance

cotillion and ballroom dance both are nouns.

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

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

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

Other hypernyms of cotillion include ballroom dancing.

cotillion (noun)	ballroom dance (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