Olympic champions win medals, while FIFA champions win trophies.
Why do the winners receive a different object as an award?
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This general trend holds across most sports, but does not generally rely on the nature of the sport itself.
The reasoning behind any particular organisation's award of either a trophy or medal (or other object) can vary, but is generally based on tradition - copied from similar events/sports/organisations or from the originators of the cup-shaped trophy most commonly used, the ancient Greek Olympic games (which awarded the winner an amphora of sacred oil).
Teams as a whole will usually receive a significant object as a trophy, while individuals receive a smaller token like a medal.
However, this is not a universal rule.
Individual players in the FIFA World Cups are given medals, in addition to the team being awarded the trophy
The individual winner of many competitions is awarded a trophy (and usually a separate smaller replica, which they keep after the main trophy is given up)
The winners of combat sport events or titles is often awarded a belt