It is the question we get asked most, and the honest answer is that it depends on where you are going and how you like to travel. For most of a slow European trip the train wins. For a few places, a car earns its keep.
Take the train between cities and along the great scenic lines. It costs less than you expect once you book a little ahead, you skip the traffic and the parking, and you spend the journey looking out of the window instead of at the road. City centres are built for people who arrive by rail.
Rent a car for the countryside, where the good places sit at the end of small lanes no bus reaches. The hill villages of Provence, the back roads of Tuscany, the quieter corners of the Dordogne. Pick the car up as you leave the city rather than at the airport on arrival, and drop it before the next one. A few days of driving in the right place beats a fortnight of it everywhere.
Our usual pattern is trains for the spine of the trip and a short car rental in the middle for the rural stretch. You get the calm of the rails and the freedom of the wheel, and you pay for neither when you do not need it.