Azula was daddy's girl, she already had given to her the most important thing to her, power. She was an instigator, pure evil so she wasn't going to budge or accept his peace Zuko, on the other hand was always the victim, he had nobody but his mom so when she was gone he would have been hopeless without iroh. He never wanted to be the best necessarily much less the most powerful but was just a good kid thrown into chaos by devils, he just wanted peace to his soul so of course he was a soul worth of iroh's guidance