country dancing vs folk dancing

country dancing and folk dancing both are nouns.

Word	Noun	Adjective	Verb	Adverb
country dancing	Yes	No	No	No
folk dancing	Yes	No	No	No

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

country dancing: a type of folk dance in which couples are arranged in sets or face one another in a line
folk dancing: a style of dancing that originated among ordinary people (not in the royal courts)

Other hypernyms of country dancing include folk dance.

country dancing (noun)	folk dancing (noun)
a type of folk dance in which couples are arranged in sets or face one another in a line	a style of dancing that originated among ordinary people (not in the royal courts)