common room vs living-room

common room and living-room both are nouns.

Word	Noun	Adjective	Verb	Adverb
common room	Yes	No	No	No
living-room	Yes	No	No	No

As nouns, living-room is a hypernym of common room; that is, living-room is a word with a broader meaning than common room:

common room: a sitting room (usually at school or university)
living-room: a room in a private house or establishment where people can sit and talk and relax

Other hypernyms of common room include front room, living room, parlor, parlour, sitting room.

common room (noun)	living-room (noun)
a sitting room (usually at school or university)	a room in a private house or establishment where people can sit and talk and relax