We always drive to Florida as a family, always have from Ohio which for us, is also a 15 hour drive. We also always drive straight through. We have done this several different ways. When the kids were younger, under 5, we would leave by midnight that way the kids are sleeping for most of the trip. In more recent years, we generally leave by 5am and we are in Orlando around 8pm. We stay off site at a Travelodge for under $50 and then we head to the resort early in the morning to check in and start playing. This way we also can get a good night's sleep before we play.
In the car, the kids watch dvd's, play travel games, sleep, listen to their mp3 players, etc.
ETA: When we stop for fuel, we use those as the potty stops as well. We are able to make some stops close to meal times so that helps to cut down the number of stops as well. When the kids were younger, we really limited their drinks and snacks in the car so that they wouldn't need to stop more often. As they have gotten older, we just make them wait! lol!