If boxers were simply violent people who wanted to beat the snot out of one another, they'd scrap in the streets. But the competition aspect of fighting is as critical as the visceral thrill of smacking another person in the face. Boxing is a sport, not just a brawl, and it involves training and practice before a big match in front of an audience, so everyone will know who wins and who loses.
Rocky would never get personal satisfaction from pummeling Apollo in a back alley, because it isn't hurting Apollo that's the goal. It's succeeding at the sport he loves.
Questions About Competition
What motivates Rocky to train as hard as he does?
Why doesn't Apollo train as hard as Rocky trains?
Is Rocky's reward worth the effort he goes through to achieve it?
Chew on This
Fame matters. The fight with Apollo is different from the fight with Spider Rico, not because of the difference in skill, but because of the difference in venue. Millions of people watch the championship fight, giving it much higher stakes.
Rocky is his own worst enemy, so he is competing more with himself than anyone else. He has to prove to himself that he can set a goal and achieve it.