baseball game vs ball game

baseball game and ball game both are nouns.

Word	Noun	Adjective	Verb	Adverb
baseball game	Yes	No	No	No
ball game	Yes	No	No	No

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

baseball game: a ball game played with a bat and ball between two teams of nine players; teams take turns at bat trying to score runs
ball game: a field game played with a ball (especially baseball)

Other hypernyms of baseball game include ballgame.

baseball game (noun)	ball game (noun)
a ball game played with a bat and ball between two teams of nine players; teams take turns at bat trying to score runs	a field game played with a ball (especially baseball)