ballgame vs ball game

ballgame and ball game both are nouns.

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

As nouns, ballgame and ball game are synonyms defined as:

ballgame and ball game: a field game played with a ball (especially baseball)

ballgame (noun)	ball game (noun)
a field game played with a ball (especially baseball)	a field game played with a ball (especially baseball)
a particular situation that is radically different from the preceding situation