bar billiards vs table game

bar billiards and table game both are nouns.

Word	Noun	Adjective	Verb	Adverb
bar billiards	Yes	No	No	No
table game	Yes	No	No	No

As nouns, table game is a hypernym of bar billiards; that is, table game is a word with a broader meaning than bar billiards:

bar billiards: a table game in which short cues are used to knock balls into holes that are guarded by wooden pegs; penalties are incurred if the pegs are knocked over
table game: a game that is played on a table

bar billiards (noun)	table game (noun)
a table game in which short cues are used to knock balls into holes that are guarded by wooden pegs; penalties are incurred if the pegs are knocked over	a game that is played on a table