In most performances of Peter Pan, the actor who plays the children's father also plays Captain Hook. In this story, the children's father is Peter Pan.
When grown-up Peter first meets Hook, he tries to show them his passport as proof of his identity. Smee grabs it away and tosses it aside. So how does he get back home? When he gets back to England, he is home.
Tinker Bell speaks English instead of her traditional ringing bell "fairy language". I always figured that the writers were ignoring the fairy language for the sake of the audience understanding her dialogue. Then it occurred to me that the only characters she talks to are Peter, Captain Hook, and the Lost Boys, all of whom could understand fairy speak in the original story. So Tinker Bell really is speaking Fairy and Translation Convention is at work. Which adds another layer to adult Peter's first meeting with Tink - for all he's "grown up" he can still understand fairies, probably without realizing he was doing it.
Jack's drawing implies that he hopes his dad falls out of the plane with no parachute. The Drawing takes on a double meaning when he finds out his father is Peter Pan.
The Pirates and Hook somehow manage to get to the real world to steal Pan's children. The reason for this is that in original adventure, Peter shanghaied Hook's ship to take Wendy, John and Michael home. To do this, Tink Fairy dusted the ship and everyone used their happy thoughts to fly it back to England. Obviously, Pan took the ship back and somewhere along the line Hook gets it back. The dust is still on it. His happy thought: One day meeting Pan again to finish the war they started.
A small one in Hook: On her way to Neverland, Tinker Bell sprinkles a kissing couple with fairy dust, causing them to float — above a bridge. The question is: Will they float just as gently back down?
Hook tries to invoke this on Peter to psyche him in battle.
Hook: You know you're not really Peter Pan, this is all a dream. When you wake up you'll be Peter Banning: a cold, selfish man, who drinks too much, is obsessed with success and runs and hides from his wife and children.
More of a Squick comical sense, but in the beginning Maggie plays Wendy in a kid's play of Peter Pan. During which much like the actual Wendy she expects a kiss from Peter instead of a thimble. It seems cute at first, but then she find's out that the person Play Pan is based off is her father....
Near the end of the film, we see Jack and Maggie fly back into the room, but with Peter, we see him asleep on the ground as it was All Just a Dream. It may be some age difference with people coming back from Neverland, and shows that Peter really has grown up.