It depends on the sport. For many sports, they are used interchangeably. For example, a football game and a football match have the same meaning. But in tennis, the words have different meanings. In tennis, points make up a game, games make up a set, and sets make up a match.

The terrible truth - in the United States at least - there appears to be no actual difference but they are not interchangeable. You simply must memorize which terms are paired with which sports. football GAME basketball GAME baseball GAME volleyball GAME soccer MATCH OR GAME tennis MATCH cricket MATCH polo MATCH A GAME of chess (casual) OR a chess MATCH (official) A GAME of golf OR a ROUND of golf