Pohnpei's previous post covers most of the bases concerning the relationship between Ponyboy and his oldest brother, Darry. Darry felt like he had to be tough on Pony (as his parents may have done) because he saw that Pony had more potential than either he or Sodapop. Darry specifically complained about Ponyboy coming home late at night, about smoking too many cigarettes and about walking alone (with rampaging Socs on the loose). Pony's belief was that Darry didn't like him as much as Soda and that he was picked on relentlessly. Of course, he was the youngest of all the characters in the novel--another reason Darry kept closer tabs on him--and his own immaturity got in the way of seeing that Darry's actions were just an example of tough love.
The main cause of this conflict, in my opinion, is the fact that Darry has been forced to serve as Ponyboy's parent or parents. Even though Darry is so young, he has to do this (and he had to give up any hopes of going to college) because their parents died.
This leads to conflict for two reasons. First, it must surely put a lot of stress on Darry. He has to try to raise two younger brothers (who aren't that much younger than him) while he works. He has to do this at an age when most people do not have much responsibility.
Second, it makes Ponyboy unhappy because Darry is his brother acting as his parent. It is hard for Pony to accept the way Darry tries to protect him and make rules for him. Darry is not that much older and yet Pony has to accept him as the boss of the family.