They are not synonyms; my best example of what I mean is thus: “Baseball is a national sport that features 160 games that are played every year between the months of April and October.” Of course, baseball is more commonly referred to as the Game of Baseball; in this sense, it is possible to have a series of games that are played within one overall game. Sport refers to the athletic event itself; games refer to the effort, governed by rules and procedure, that give definition to that event. The sport of baseball is different from, say, the sport of football, in that each requires a certain unique set of athletic abilities. A typical game of baseball is divided into (nine) innings; (four) quarters divide a game of football, and so on. In English, words that sound as if they are the same don’t necessarily have the same meanings.