Is there a price difference purchasing train tickets from the internet in the U.S. than purchasing in Italy?
The only advantage is the opportunity to purchase nonrefundable SuperEconomy and Economy tickets for future trips here at home, rather than waiting until you are there and most of these fares will be sold out for the immediate future. Then only full fare (refundable) tickets are available.
For long distance high-speed trains, booking in advance can save you a lot of money. For example, the base (standard) 2nd class fare for Rome to Venice is €80.00. The Super Economy fare is €29.00. On some routes the Super Economy fare is €19.00. There is no advantage to buying tickets in advance for travel on regional trains. Tickets are cheap and there are no discount fares.
Tillie, As the others have mentioned, if you pre-purchase tickets on the premium trains such as the high speed Freccia trains (which have compulsory reservations), you can save money. The reality is that the Super Economy tickets usually sell out very quickly, but you may be able to get Economy fare which will still provide some savings. If all that's left is Base fare, it doesn't matter whether you purchase in advance or not. If that's all that's left, it's just as easy to buy those in Italy. Note that tickets for premium trains are specific to a particular date, train and departure time and can ONLY be used on that train. If you board another train by mistake, you'll face hefty fines. As Tim mentioned, purchase of tickets for Regionale trains is not discounted, so just buy those in Italy from a staffed ticket office or one of the Kiosks. They usually can't be purchased more than seven days from travel date. One IMPORTANT point is that when using Regionale trains, the tickets MUST be validated prior to boarding the train on the day of travel, or you'll risk hefty fines which will be collected on the spot! You may find it helpful to have a look at the "Transportation" section of the excellent www.roninrome.com website. Happy travels!