ball game vs ballgame

ball game and ballgame both are nouns.

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

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

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

ball game (noun)	ballgame (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