football game vs field game

football game and field game both are nouns.

Word	Noun	Adjective	Verb	Adverb
football game	Yes	No	No	No
field game	Yes	No	No	No

As nouns, field game is a hypernym of football game; that is, field game is a word with a broader meaning than football game:

football game: any of various games played with a ball (round or oval) in which two teams try to kick or carry or propel the ball into each other's goal
field game: an outdoor game played on a field of specified dimensions

Other hypernyms of football game include contact sport.

football game (noun)	field game (noun)
any of various games played with a ball (round or oval) in which two teams try to kick or carry or propel the ball into each other's goal	an outdoor game played on a field of specified dimensions