pas de deux vs dancing

pas de deux and dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
pas de deux	Yes	No	No	No
dancing	Yes	No	No	No

As nouns, dancing is a hypernym of pas de deux; that is, dancing is a word with a broader meaning than pas de deux:

pas de deux: (ballet) a dance for two people (usually a ballerina and a danseur noble)
dancing: taking a series of rhythmical steps (and movements) in time to music

Other hypernyms of pas de deux include dance, saltation, terpsichore.

pas de deux (noun)	dancing (noun)
(ballet) a dance for two people (usually a ballerina and a danseur noble)	taking a series of rhythmical steps (and movements) in time to music