We are debating this right now and have done both in the past. We have gone the last two Augusts. Last year we drove down from just north of Baltimore. We left about 8pm and drove straight through the night. We arrived in Disney about 11:30 am. We stopped for breakfast in GA for about 1.5 hours and maybe another hour for gas stops and breaks overnight. We also ran into I-95 road construction traffic in VA that slowed us down for about 2 hours. We made awesome time with only about 13 hrs of drive time even with the traffic in VA. We weren't so lucky on the way back. Terrible rain storms and traffic made us have to stop overnight in SC. Our drive home was much longer, more like 16-17 hours.
We are trying to decide between flying or driving this summer. Last year our costs for driving were about $300 in gas (minivan), $125 hotel room, $250 food costs, so about $700. I can get flights right now out of DC (as it is cheaper than flying out of Baltimore) for about $950 for 5 of us. I am thinking the $250 more is worth the hassle of driving 30+ hours round trip. Only downside, I hate taking the busses and would want to rent a car while we are there.
My husband and I have a long history of doing long drives (he was in a band before we had kids and we toured so we have done lots of long overnight drives). Driving isn't for everyone. Our kids are also very good with driving. Last year they were 5,3 & 8. They had no problem sleeping in the van and pretty much all night. My husband took off the Friday we left and slept a good portion of the day so he was good to drive through the night. It was cake on the drive down, but like someone said the drive home sucks...there isn't anything to look forward too.
We are also considering the Auto Train. It is more expensive, however, you save your gas costs and the wear on your car. I also thought it might be fun for my kids.