dictionary.com defines a game as "a competitive activity involving skill, chance, or endurance on the part of two or more persons who play according to a set of rules, usually for their own amusement or for that of spectators."
Sport is defined as "an athletic activity requiring skill or physical prowess and often of a competitive nature."
There is virtually no difference in the meaning of them, even when using them as verbs. For example, you can make fun of someone by making game of them, or sport of them.
Where the two words differ somewhat is when they are used as adjectives. For example, you hunt game (used as a description of what you are hunting). Sport as an adjective can describe any number of games and is also used to describe clothes.