I have driven once from the DC area (about 12 hours - but now that there is construction on the beltway south of DC I wouldn't do it anytime soon again).
That being said, my kids were 2 and 4, and we left after lunch, and they were asleep by 8pm, so it was pretty smooth sailing. We were gone for 2 weeks (11 nights at Disney, then went down to Naples to see my folks, then a two day drive home, which sucked royally). We were loaded down in our minivan, had movies to watch, and it was really comfortable.
Our reason for driving was that we had just gotten our first minivan, and it was great, and it was after 9/11 so we were not feeling like flying that trip. It wasn't financial, just wanted to do it (and brought all our food and drinks since we were staying at the Boardwalk Villas, as well as double stroller and stuff, so it was nice to fill the back of the van, matter of fact no more than 4 would have squeezed in with all our stuff!).
I would say if you were going for two weeks or so it may make more sense to drive, but for a week and trying to have the energy to make early breakfast reservations after such a long drive, you may be wiped out at first. Plus, will you have enough room for all your stuff in one car?
My kids have flown a lot since they were babies, and it is work to fly, but once they were 5 or so they could pull their own carry on bags in the airport so it got easier to travel with them! It's hard with babies and seats and all, but it shouldn't stop anyone!
I would price out flights. It's such a difference to spend just the morning traveling (of course getting to the airport in time, we have breakfast once through security, waiting for the flight, etc... but still then in two hours you're in Orlando). That versus 14 hours plus stopping in a car.
We've driven to St. Louis several times - also 12 hours. If it's all 4 of us we can do it in one LONG day. I did it alone with my kids a few years ago, but took two days (and it was still LONG). Stayed two weeks or so.
Only you know your budget, but I would look into flying. If you do drive, plan plan plan. Take breaks. Stop and have bathroom breaks, let everyone run around to stretch their legs, then eat in the car to save that time. It is doable but takes some planning!
Good luck with your decision.