Lots of emulating in sport too. It depends on the situation, group dynamics, etc.
emulation: ambition to equal or excel
competition: strive to gain or win something by defeating or establishing superiority over others who are trying to do the same.
As I climbed a five pitch route, it is impossible that one climber stay on the ground and the other climb the pitch three. For that reason, you are looking for some one who equal or excel your ability and will try to be better.
In you sport climb, you don't need some one who can follow you or lead you to the top. You can work a route and if some one else did it, you can stay for hours in a cliff to beat him. In fact, the most competitive climber just need some one to hold the rope as he climb.
As a matter of fact, in sport, group are easier to make. Many people is at the bottom looking at you ready to laught if you mist it or applaud if you are better. In trad, the group is two persons. You just have to beat your partner if you compete. or you can work as a team where each people teach at the other some think. Even modesty by a newbye.
In trad, people will seat down in front of cathedral looking at climber and discussing technique as they take a beer. In sport they don't even think to look at a climber in refuse, except to notice there mistake.
It is little difference and there is more, but I think that they exist
Even in a forum, you read competitive note.