On a regional train, if they have 2 classes, the difference is not worth the money. Maybe it gets you a slightly cleaner seat, slightly. I think it's more important to get a reserved seat than a first class seat.

To be specific, this is the route I am looking at for April 30th:

11:12 a.m. Roma to Perugia (through Cortona): 2 regional trains.

Trenitalia is showing two classes of service on these regional trains: First is E18,30, second is E11,70 (don't know how to make the euro sign).

Maybe first-class is one of those glassed-in compartments with a door?

I wish we could make the 9:38 a.m. Eurostar from Roma to Perugia (through Foligno on a regional train), but we have no wiggle room even if we land on time at 7:35 a.m. at FCO.

The regular, non-first class cabins also have the glass compartments. It just depends on the train.

Perhaps 1st cl has AC and 2nd doesn't. I don't think you can expect much on R trains whether you buy 1st or 2nd. Not all R trains offer a class selection.

First class seats are a little bigger, with more leg room. Typically, six first class seats take the same space as eight second class.
The first class part of the train may be quieter and you might rub shoulders with a better class of person.
The railways provide first-class seats because there are people who want to pay the extra fares.
I've travelled all over Italy by train in second class, and have never considered paying the extra for first class.

It can depend on the equipment. On some regionals there may be one or two carriages which have a First Class section in sndwiched inbetween second Class sections at each end.

Believe me, they are NOT air conditioned in all cases and the seats aren't any cleaner, or anything else much different...