"Hook or me this time."
With Part One having prepared us for the dark side of Neverland, Peter's oath ends chapter 13 and sets up the controlling dichotomy for the last third of the book. It's more than just the two characters, though, more than just Hook vs. Pan. It's a too serious adulthood against a too forgetful childhood. They are extremes, to be sure, and in the middle is Wendy and all the rest of us.
For most of the book, James Hook is the perfect villain, the best pirate. He's erudite, articulate (even when he swears) and lethal. There are some brilliant fights with the lost boys and when he finally captures them, he breaks the Neverland rules to do so. His charmingly sinister nature sets the mold for characters like Jack Sparrow. This brilliance can't last, however, since he is an adult in a child's world.
After capturing Wendy and the lost boys, and nearly poisoning Peter in the process, Hook becomes obsessed with his lack of "good form." We find out "he had been at a famous public school; and its traditions still clung to him like garments." Even as Peter launches the rescue and final battle, Hook's past torments him. He can't let it go, especially when he realizes Peter has better form than he does. In the end, he stops fighting for his life and waits for Peter to commit a small infraction. When this happens (a kick where there should have been a stab), he triumphantly leaps into the waiting jaws of the crocodile.
Peter, for his part, seems to have no clue about Hook's obsession. Indeed, Peter is relatively clueless for the entire story. If it doesn't have to do with the adventure of the moment, it's meaningless to him. The past and the future are non-existent to the Pan; all that matters is what he is doing now. Nowhere is this made most apparent than towards the end when Wendy tries to reminisce with Peter about the battle with Hook. Peter looks at her with interest and asks "Who is Captain Hook?" The great nemesis is gone, from the Neverland and from Peter's mind. He even forgets Tinker Bell after she dies of fairy-old age. Whereas Hook was undone by the past he couldn't forget, Peter forgets everything and so is bound by nothing. Perhaps that's why people love Peter Pan so much. It's not because we all want to be young again, necessarily, it's that we want to escape the responsibilities which bind us and hold us down. Guilt comes of thinking beyond ourselves, beyond the now, two things which Peter Pan simply cannot do.