table game vs billiards

table game and billiards both are nouns.

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

As nouns, billiards is a hyponym of table game; that is, billiards is a word with a more specific, narrower meaning than table game:

table game: a game that is played on a table
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 (noun)	billiards (noun)
a game that is played on a table	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