billiards vs table game

billiards and table game both are nouns.

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

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

billiards: any of several games played on rectangular cloth-covered table (with cushioned edges) in which long tapering cue sticks are used to propel ivory (or composition) balls
table game: a game that is played on a table

billiards (noun)	table game (noun)
any of several games played on rectangular cloth-covered table (with cushioned edges) in which long tapering cue sticks are used to propel ivory (or composition) balls	a game that is played on a table