Backing up Tim here: buying train tickets for arrival day is always a risk as you have to hope your flight is on time and there aren't any other delays, like holdups in baggage or a very long immigration line. This really only applies to the super-cheap, super-economy fares, though, as they're non-refundable and non-changeable. Sometimes these are SO inexpensive that it's OK to take the risk as long as you are willing to eat the price, if necessary.
Economy tickets can be changed but you have to pay the difference between what you paid for them and the price on the day of the journey.
Full-fare tickets can be changed at no cost.
What time are you getting into Fiumicino? There are two trains a day which go directly to Florence (S.M. Novella) from there - at 11:08 and 15:08 - so you might not have to take the express to Rome if you can catch that first one. That would be a cost saving of 56 euro (14 euro apiece for LE tickets) right there.