On APs vs sandbox, I don't think it's my job to explain how they use different techniques: it's not a comparison or contrast I'm terribly interested in.
If I had to start I would say that a sandbox consists of events on a trajectory, like
@Emerikol's example upthread of the murder, and the expectation of play is that the players will declare actions that prompt the GM to change the trajectory.
Whereas an AP consists of events on a trajectory, and the expectation of play is that the players will declare actions that conform to the established trajectory.