Combat suggests a direct encounter, one actually involving physical force: "The FBI is supposed to combat crime." "The fight between the boxers was a frightful, bloody combat." Contest applies to either a friendly or a hostile struggle for a prize or goal. A baseball game is a contest; a battle between armed forces is both contest and combat. Contest should be associated with struggle and contention; combat should be related to conflict and fight.
Dictionary of problem words and expressions. Harry Shaw. 1975.