ballgame vs baseball

ballgame and baseball both are nouns.

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

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

ballgame: 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 ballgame include ball, baseball game.

ballgame (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 particular situation that is radically different from the preceding situation	a ball used in playing baseball