ほとんど同じですがcontestを動詞として使うときには反対するという意味もあります。competitionをそうゆうふうには使えません。例えば、We contested his decision　を言えますが、We competed his decision とは言えません。名詞としてはWe held a competition とWe held a contestがほとんどいっしょうです。もしかしたら、competitionのほうがちょっぴり丁寧な表現といえるかもしりません。

A competition is described as a contest between individuals or teams to decide a winner. On the other hand, the word contest is used to describe a competition between participants that gather to win an event. Read more: http://www.differencebetween.com/difference-between-contest-and-vs-competition/#ixzz2UTjIHON0