L.R.
Tara is right about the fact that if your kids need a break from being on the move, you cannot do that on a train, nor can you stop if a kid feels ill. Good points. A lot depends too on whether you make the drive itself part of the vacation, stopping at interesting places en route. You can't pop offf the train at stations and get back on. And please, do bring a TON for your kids to do. I remember a five-hour train ride in a seat across from a mom with three kids and she had nothing at all for them to do, see, read, no paper to draw on -- nothing. Of course they were a handful.
I haven't done that exact route (Disney) or a trip that long on Amtrak but I have taken Amtrak a lot, all my life. It's a great way to travel and lets you read or play games etc. or even work rather than focusing just on the road. But I would say that with kids you should be aware of the following:
You can get sleeper car berths so you and the kids get actual beds. Amtrak seats do not recline much at all and I would not want to spend an entire night sleeping in one. Kids might be able to do it, but as an adult I'd be a mess if I tried to sleep in one as my "real" overnight sleep. Also, you will be awakened every time the train stops, or every time someone in the seats around you decides to wake up and have a loud phone conversation at a ridiculous hour. Yes, that does happen.
Amtrak Superliner trains offer a "family compartment" that sleeps two adults and two children -- check their web site. It could be a real adventure for your kids! Compartments are private, just for your family, though the family compartments do not have private toilets (other compartments sometimes do). If your kids have issues with using a toilet that smells and has a wet floor, you will have problems -- I say this as an Amtrak fan: The shared toilets usually stink not long after the train starts and keep on stinking.
Another option is the Amtrak Auto Train -- you can take your own car ON the train -- but you would have to drive all the way to the Washington, D.C. area to board that train, which runs only between Lorton, Virginia, to Sanford, Florida. But it does at least enable you to have your own vehicle with you IF that matters and it might not.
I would think you'd do better to get a family compartment all the way from Boston to Florida. But you might find the cost ends up being as much as hotels on the road would be. The trade-off is that you don't have the hassle of driving that very long drive!
Have you checked out how to get from the destination train station to the Disney property you're considering? Does Disney run shuttles or are taxis readily available and is the distance reasonable so it's not expensive in a taxi? Something to consider too. You don't want to take the train only to find that it's a massively expensive cab ride or shuttle ride all the way to your destination.