ball game vs baseball

ball game and baseball both are nouns.

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

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

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

Other hyponyms of ball game include ball, baseball game.

ball game (noun)	baseball (noun)
a field game played with a ball (especially baseball)	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 ball used in playing baseball