I have traveled often on Amtrak and highly recommend it. You and your children are able to get up and move around. You can take snacks/a meal and eat when you want or you can go to the snack/dining car. Personally, I recommend taking your own food. The dining car can be expensive tho, in my experience the food has been good. I've done it both ways. I avoid the snack car. It is expensive and sells the little packaged foods that are easy to take with you.
Riley's list of how comfortable it is is a good list. I've never purchased my own cabin and always sat in general seating. I've found general seating to be comfortable even when it's crowed. You will have less room but you can still get up and move around.
I don't know why driving would not be safe. I've driven cross country and back. Seven hours is not long at all even with kids as long as you take along toys and things for them to do. My granddaughter, at around age 2, and I drove several times for 3-4 hour stretches and she was fine. We did stop a couple of times so that she could get out and run around. And we did stop to eat.
Now a days you can have a cell phone and call for help if you do have car trouble. Several years ago my daughter and I had to stop in an isolated area at night because of car trouble and had to rely on a stranger to see us and stop. He called AAA for us. It turned into an adventure.
You don't say how old your children are or how well they get along. That would make a difference. However, if I had the choice between driving and taking Amtrak with 2 children I would take Amtrak. I recently went to Seattle with a friend and we took Amtrak and enjoyed it.