The best option may be a mix of the two. A car is useless for most of your listed destinations since they are all cities that do not permit the use of cars in their city centers unless you are a resident with a permit. However if you intend to do some roaming around the countryside and the small towns in Umbria while in Assisi, then a car might be useful only while in Assisi.
But aside from that, for all other locations you would be spending an average of at least $400/week to rent a car which would be sitting in the hotel garage all of the time, except for the time to transfer from city to city. The cost of the rental itself (which varies but should be at least $400/week) is the major cost of the car. Gasoline is expensive in Italy (a bit less than 2€ a liter, or almost $8/US gallon), however for your trip the distance traveled should not be more than 1500km in total (and I’m exaggerating) therefore about 100 liters of fuel (€200) plus another €100 for tolls max. So your car cost for 3 weeks should be less than $2,000 including rent, gas, tolls, and parking fees which you will incur in many cities and possibly even at the hotels where you choose to stay, I’m not taking into consideration traffic fines, because I presume you will observe the speed limits, will not drive on bus lanes, will pay for the parking meters, and won’t drive inside traffic restricted zones (ZTL).
Train cost, even if you travel only on the fastest trains (where available), at full base price, without taking advantage of discounts with pre-purchase, should not exceed €800 in total for both of you.
So if you are looking to save the train is definitely the cheaper option for two folks.