Olympic champions win medals, while FIFA champions win trophies.
Why do the winners receive a different object as an award?
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