baseball game vs ballgame

baseball game and ballgame both are nouns.

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

As nouns, ballgame is a hypernym of baseball game; that is, ballgame 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
ballgame: a field game played with a ball (especially baseball)

Other hypernyms of baseball game include ball game.

baseball game (noun)	ballgame (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)
	a particular situation that is radically different from the preceding situation