There are many sports and other events that are contested, but why are some contests called matches, like tennis match, golf match, and soccer match, and some contests called a game, like baseball game and football game? It's not a baseball match, or a tennis game, so is there a rule of English that differentiates a match from a game?.
"Match" sports tend to be the ones that originated in the UK or other Commonwealth countries. Consider:
The sport where players kick a round ball around a field is a bit of a special case: it's a football match in the UK, but a soccer game in the US. Since the US didn't import the name, apparently it didn't import the "match" terminology either, and the homegrown "game" tradition got applied to it.
A "game" is one unit of play according to set rules in which a winner can be decided. This is distinct from a "point" or "round", which is a possible substructure of a game in which one player may "win" or "lose" standing in a single game due to the scoring or winning of points or rounds, but has not won or lost the game itself.
A "match" is an "event" consisting of the "matching" of two (or more) opponents or teams in a competition involving the playing of one or more "games". If the match consists of more than one game, additional rules provide for the selection of an overall winner based on the number of games won by each player. A "game" of tennis is decided by the first to win at least five "points" with a margin of two, where a tennis "match" is decided by a more complex structure divided into "sets" which consist of "games". However, as noted in the comments, a single "game" of soccer (football) also constitutes a "match"; the two teams meet up and play one game, and that is the "event"; the terms in this context are thus interchangeable.
A "tournament" is a competition structure, often recursive in nature, consisting of many "matches" between subsets of competitors (usually but not always pairs), where the winner of each match gains standing and the loser loses standing among competitors in the overall structure.
A "contest" is overarching: it is any competition between two or more parties where the goal is to demonstrate superiority, which includes but is not limited to "games" (war, sweepstakes, lotteries etc. are also "contests" but do not meet other criteria of a "game").
See the Wikipedia article on "game" for more attempts to differentiate between terms for various leisure activities:
Further, a game is a sport if one of its main properties is a high level of physical exertion or full-body coordination.
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The difference here is just cultural convention. Different sports had roots in different time periods and places, and tend to keep the terms popular at the time.
Also some of it is arbitrary. In tennis, for example, a player needs a certain number of points to win a game, six games to win a set and three sets to win a match. The hierarchy is pretty much artificial.