Contest refers to an organized event such as a singing contest with more than a few participants, e.g. the supergirl contest.Competition refers to the process between or amongst participants for the final outcomes such as the championship. I just made this sentence as illustration:魏佳庆 and 李璇 are competing for the championship in the 沈阳 contest.
contest多指不同意某个观点而争辩。例如 I must contest. I think you are wrong.competition泛指为了取得某方面的胜利而进行的比赛、竞争等。例如He was in competition with 10 others for the job. He takes part in a swimming competition.