Train Tickets

If you are planning to take any of the high speed Freccia trains, yes as you can get discounts in advance if you are willing to commit to date and time as they are non-refundable. What is your itinerary?

The only real advantage to purchasing ahead is to save money on the high-speed trains, and if you're going in just a couple of weeks, there probably won't be bargains to be had. In any case, if you get one of the bargain tickets, it locks you into a particular train and departure time - you certainly wouldn't want to get one for your arrival day, in case your plane is late. There is no point buying the Regionale train tickets in advance, since they're all priced the same and never sell out.

there probably won't be bargains to be had

I wouldn't bet on it.

Trenitalia has "Economy" discounted fares on sale for the Milan-Rome high speed trains departing tomorrow morning at 7:20 AM, 8:20 AM, 11:20 AM and so on.
It's Not a big discount, 79.9 Euro per person vs a Base fare of 92, nevertheless discounted tickets are still available less than 24 hours before the departure.

Many Europeans do not purchase train tickets in advance, those travelling on main routes still enter train stations without having checked the timetable at home.

There is no point buying the Regionale train tickets in advance

I agree, but Regionale tickets purchased on-line do not need to be time-stamped before boarding. It may be a good reason for tourists not used to open tickets.

Any ticket you buy for any of the trains other than the Regionale trains locks you into a specific train, date and time and can only be used for that specific train. There are three levels of ticket pricing in Italy --- Super Economy, Econ, and Base. The first two are discounted tickets and come with restrictions related to change and refund. The Base is the walk up fare and is changeable until an hour after the train has left the station. All of those tickets come with a specific seat reservation. Only the Regionale trains tickets are open and can be used on any Regionale train headed in the correct direction for four hours after it has been validated (time stamped).

If you are worried about the train being sold out, that rarely ever happens. Trains are too big and too frequent.