A legal battle is more a contest than a competition. If Fred sues Bob in court, then Fred and Bob would be called "adversaries", not "competitors". I like Elijah's example of "sibling rivalry" as a competition, not a contest. Siblings are competitors, not adversaries. If their rivalry became a contest where one child were to be declared the winner, the family would be on the verge of breaking apart.

I just thought of another example that could be helpful: the Olympics. That's not a contest, it's a competition. Each activity (marathon, shot put, boxing match) might be called either a contest or a competition without much difference in the terms, but I wouldn't call the Olympics a "contest" but a "competition."

‘Competition’ suggests rivalry in a more direct way than ‘contest’. For example: The zoo held a contest to name the newly-born lion cub. (The point isn’t that someone wins, but rather that the lion ends up with a popular name) The best-selling author sponsors a poetry contest for students in his home town. His goal is to stimulate interest in poetry and create a community of people interested in poetry, not to determine the best poet. To qualify for the State finals, a gymnast must place in the top five of one of the regional competitions. (Their purpose is to select among the competitors.)

All contests are competitions but not all competitions are contests. In the majority of situations, you can use either term. You’ll rarely be wrong. However, to answer your question, competition is a broader term than contest. In a contest, a person or team competes for a prize (or recognition). Personally, I might prefer this term if only one skill were being tested, like a pie eating contest, although competition also works well in that context. Imagine, however, a broader concept, less precise regarding the specific skill being tested. An office, school or family might be a competitive environment, and the workers, students or siblings might feel like they’re in a competition. I probably wouldn’t use contest in that context. Again, I’d say there’s very little practical difference in most situations.

When describing an event, they are the same. Sports games, eating challenges, poetry jams, etc could be called either contests or competitions. But competition can also be considered an emotion. Sibling rivalry is competition but not a contest. Two big predators, like a puma and a wolf, in the same area might have a lot of competition for food, but it isn't a contest. Having written this out, I think duration may play a role. A one-time event where participants are doing something in order to win over other participants is a contest or a competition, but something on-going, without clear rules or boundaries, is a competition without being a contest.