football game vs contact sport

football game and contact sport both are nouns.

Word	Noun	Adjective	Verb	Adverb
football game	Yes	No	No	No
contact sport	Yes	No	No	No

As nouns, contact sport is a hypernym of football game; that is, contact sport 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
contact sport: a sport that necessarily involves body contact between opposing players

Other hypernyms of football game include field game.

football game (noun)	contact sport (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	a sport that necessarily involves body contact between opposing players