Traveling from Provence to Amboise

It will likely take about the same time, since the train trip will require changes. Travel by train might be cheaper, depending on the type of ticket you buy. And may be more relaxing. Plus you don't have to be concerned with gas, tolls, photo radar, and traffic slow downs. OTOH, there's no doubt that having a car to explore the Loire is advantageous.

Best of both worlds would be to take an early enough train to arrive in St Pierre des Corps while the rental companies are still open and rent your car there.

From where in Provence?

We're traveling from Arles to Amboise, at least that's what we've planned.

This is always going to be a long day by public transport. Not only is it a long way, but both are relatively insignificant towns not terribly well served by anything but local trains. (I like both towns more or less, but if I was travelling to either I wouldn't start from the other...)

Add to the mix the fact that however you do it by train you're going to have to get yourself across Paris.

I would look at travelling Arles to Paris, staying a couple of nights, then going on to Amboise. Once in Amboise you'll be kinda stuck without a car - there are transport links commensurate with a very small town - which Amboise is.