Pros:
Can visit out of the way sites
More flexible travel schedule
More lodging options - agriturismo, country villas etc

Cons:
Not super practical within major cities
Can be less relaxing than taking a train
Traffic is an issue in big places (Rome) and small (Sorrento)

Other:
Don't leave valuables in the car
Many car models have little luggage space
If you're keeping a car the whole trip, definitely check on parking at the places you're staying