board game vs go game

board game and go game both are nouns.

Word	Noun	Adjective	Verb	Adverb
board game	Yes	No	No	No
go game	Yes	No	No	No

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

board game: a game played on a specially designed board
go game: a board game for two players who place counters on a grid; the object is to surround and so capture the opponent's counters

board game (noun)	go game (noun)
a game played on a specially designed board	a board game for two players who place counters on a grid; the object is to surround and so capture the opponent's counters