The Ship's Compass is one of the three items required to repair the contraption that opens the door leading to Tortooga.
The heroic path is doing the thing that Pat prefers: Mickey will have to get his treasure from the tree house, since Pat is physically unable to do it himself. After getting his treasure (which turns out to be a teddy bear named Treasure), Pat hands over the compass.
The scrapper path is simply buying the compass from the pirate for 25 E-Tickets. However, Pat will still be satisfied, claiming that the compass was actually worth no more than 24 E-Tickets.
After Smee fixes the contraption, the compass will be in its place (on some sort of box that's part of the contraption with the compass' lid flipping open and closed) for the rest of the game.