"Game" is a narrow concept while "Sports" is a broader concept.
Game includes taking part in a particular activity which gives entertainment and improves the physical skills of a person. Athletics(running, jumping and throwing), cricket, football, golf, all of these are games. Even "gili danda", a local Indian activity which provides entertainment is a game.
Sports includes taking part in such activities which not only improves the physical skill of a person but also helps in maintaining one's health. Sports includes playing of various games, taking a healthy diet, and taking part in all such activities which improves the mental skills of a person besides improving his physique.
All games are a part of sporting activities in the sense that it helps improve a person's physique, but Sports is not a part of games as Sports includes improving both physical and mental ability of a person.