I've driven almost as far as you're talking about 3 times over the past year, from the Charlotte, NC area to Las Vegas twice (one way last year, round trip in the spring) and last week I drove from the Charlotte area to Tucson, AZ. Both my kids moved west, so I made one way trips driving with them out west, then flew back, then my wife, dogs and I did a week in Vegas, driving out and back, in the spring.
If you ask me, driving across the country is well worth it. There's no way to actually see how much the country varies any other way. I guess train might be similar, but you can't stop by a Route 66 diner for an awesome meal in a train. It is true that some of the scenery is less than spectacular in some areas (lots of corn and other fields, depending on your route). For my money, though, New Mexico, Arizona & Nevada have some of the most interesting and beautiful scenery you'll ever see.
That said, here are my thoughts from my recent trips.
- From here to AZ/NV for us is 3 hardcore long days of driving. For you, I'd figure on at least 4 days on each end of the trip.
- You do make good time out west. My favorite welcome signs are the ones that say 75 mph in TX, NM & AZ. Coming back east I did a lot of eye rolling as the speeds drop back down to 65 and 70.
- Seriously consider renting a car. We rented from Hertz, when we drove to/from Vegas. They truly have unlimited mileage per week and over the course of our trip we did over 4,500 miles. It's nice to plop down in someone else's vehicle and put all those miles on and not have to worry about oil changes and the wear you're putting on your own vehicle. For us, the $300 we spent for the rental and fuel for a full size vehicle was less than half what flights would have cost.
- The biggest downside for me, on those trips, is driving by all the cool places knowing I had an end destination I had to make. For example, I've now driven by or near the grand canyon 4 times without having time to stop. I'll be making a trip there soon.

- I do have to say we were pretty darn beat after our drive home from Vegas. I'd imagine a drive home after a week at Disney would be similar. Our time was tight though and we were doing some very long days. If you have the time to take easier days, you might be better off.
Bottom line, if you're family is cool with the drive from Jersey to Florida, I wouldn't shy away from the drive. It would be a great thing for your kids (and you) to see the sights. At least that's my 2 cents.