carol vs song

carol and song both are nouns.

Word	Noun	Adjective	Verb	Adverb
carol	Yes	No	Yes	No
song	Yes	No	No	No

As nouns, song is a hypernym of carol; that is, song is a word with a broader meaning than carol:

Other hypernyms of carol include strain.

carol (noun)	song (noun)
a joyful song (usually celebrating the birth of Christ)	the act of singing
joyful religious song celebrating the birth of Christ	a short musical composition with words
	the characteristic sound produced by a bird
	a distinctive or characteristic sound
	a very small sum

carol (verb)	song (verb)
sing carols