A full car is cost competitive with planes and trains, and gives you flexibility to stop and see things along the way.
For distances where trains make sense, cars are preferable: You can travel at your own schedule, and then you have transportation at your destination. If you take a train, you have to meet the train at a certain place at a certain time, which is stressful, and then you still have to arrange transportation from the train station to your final destination.
Even worse, if you have to change trains during your trip... It's much, much easier to put all your stuff in a car once.
This basically leaves trains for traveling between dense cities, where the passengers either live close to a train station, or have a destination close to a train station.
And, to be honest, after flying twice with young kids this year, I put my foot down and told my wife road trips only for the next few years. I wouldn't want to travel by train with young children for the same reason that travel by air with young children is difficult.