Not to quibble, but "Tivoli" is actually called the "Villa D'Este". It's in the same town, Tivoli, that has Hadrian's Villa, another out-of-town Rome attraction. Both can be seen by bus or train/local-bus, but are perhaps more convenient by car. I agree that Orvieto is very attractive, but perhaps not far enough along the trip to break it in half. The superhighway in Tuscany is only marginally more attractive than other superhighways. I would not take the car unless you have use for it after Tuscany.
If you are staying in Positano, you should make sure you have an affordable parking place for the car. I suppose traffic in Capagnia is better in April, but even by the last week in May, it is fierce.
Edit: Don't forget to make a schedule for your last day in Positano. You cannot (for example) reliably make a morning international flight in Rome, having slept in Positano.