Maybe..."tournament" is not always exactly a contest. For example, they have "soccer tournaments" but I wouldn't say that a soccer game is exactly a contest. I think of a contest as more of an individual thing. In poker tournaments and soccer tournaments, there are many individuals competing against one another...but then there are "poster contests" and not "poster tournaments" and I am not sure how I would explain that.
All tournaments are contests. Not all contests are tournaments. Tournaments are likely to have several contests. Contests do not have several tournaments. Golf tournaments come to mind where each foursome is having a contest. The American football contest called the Rose Bowl is actually named the Tournament of Roses. it is a misnomer but gives a stature to the event.