The main difficulty they have begins when one of them has to be chosen as chief and the other has to lose. Once Ralph is chosen, Jack immediately feels angry about it and Ralph tries to placate him by giving him a lesser command, that of the hunters.
But they have a fundamentally different view of how things ought to go on the island and what is important. Ralph is tied to the idea of having organization, building shelters, keeping a signal fire lit, and hoping for a rescue.
Jack is bent on hunting for meat, managing their fear through hunting and violence, and "having fun" and not being concerned about a rescue. Eventually the thirst for violence and savagery becomes more acute until the conflict between Ralph and Jack and their followers actually becomes violent.
Because of the incompatibility of their two visions, there is a contest for leadership and eventually Jack wins the contest. Ralph's followers dwindle in number, then in a frenzy at the "party" Simon is killed. Once Jack and the hunters steal Piggy's glasses they have control over fire and the victory is complete. Symbolically the destruction of civilization is also complete when Roger kills Piggy and the conch is smashed.