If you're talking about an event where people compete against one another, there is not much difference. If it's a sporting event, "competition" is a little more common. If it refers to a skill — archery, graphic design — contest is a little more common. The difference is small, though. However, "competition" has an extra meaning — it can refer to the *condition* of competing. For instance, "there is intense competition for business between these two stores." Here, "competition" is not a specific event, but an atmosphere of competing.