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