"Sport" is that which is visited upon game animals by a human armed with a device of some sort used to end the pursuit of said animal.
"Game" is either a generic name applied to said animals, or something that requires sticks, balls, score keepers, and written rules. People "play" games, "sport" is not "played".
Tip O the pen to Stephen Bodio.
Best,
Ted
Unfortunately the standard dictionary definition does not agree with that. The terms are interchangeable for all intents and purposes.
The below is not as extensive as Webster but has the essence
sport
noun
an individual or group activity pursued for exercise or pleasure, often involving the testing of physical capabilities and taking the form of a competitive game such as football, tennis, etc
such activities considered collectively
any particular pastime indulged in for pleasure
the pleasure derived from a pastime, esp hunting, shooting, or fishingwe had good sport today
playful or good-humoured jokingto say a thing in sport
derisive mockery or the object of such mockeryto make sport of someone
someone or something that is controlled by external influencesthe sport of fate
informal (sometimes qualified by good, bad, etc) a person who reacts cheerfully in the face of adversity, esp a good loser
informal a person noted for being scrupulously fair and abiding by the rules of a game
informal a person who leads a merry existence, esp a gamblerhe's a bit of a sport
Australian and NZ informal a form of address used esp between males
biology
an animal or plant that differs conspicuously in one or more aspects from other organisms of the same species, usually because of a mutation
an anomalous characteristic of such an organism
