Need train from Rome to Florence.

Only if you pay full fare, but since it is very unlikely that the train will sell out unless it is something like a 3-day Italian holiday, you could just wait and buy it at the station when you want to go. The trains to Florence go about every half hour. If you change a full fare reservation, you will have to wait in the ticket line anyway to make the change.
You'll have to wade through the terms and conditions at the trenitalia site. You may be able to change an Economy ticket in advance, but then pay the added cost for full fare on the new ticket. A Super Economy ticket is completely unchangeable.

Thank you, Sam! That is great info!

Here is some of that verbiage on the Economy ticket from Trenitalia.

The economy ticket

Looking for saving? The Economy ticket is for you!

Until 30 September 2015 the ticket is available up to train
departure, on Frecciarossa and Frecciargento, in 1st and 2nd class and
for Business, Premium and Standard service, and couchettes and Wagon
Lits.

How and when I can manage my ticket?

You can change date and hour once up to departure, choosing a journey
on same category of train, according to the ticket change procedure
and paying the difference between the full price (Base) applied to the
train used and the sum already paid.

If you bought your ticket online you can change it on this website
following these steps: •displays the trip to edit in your private
area (for purchases with registration) or in the section "change
ticket" (for purchases without registration) •select the option
"Refund /Other changes" •select the travellers for which you want to
make the change and choose the "Change ticket" choose the new date and
/ or time of departure and click "continue" •choose the train on which
to travel and click on "select", decide whether to choose the seat or
not and click "continue" •pay the ticket

You can change the ticket with a BASE fare, paying the price
difference. The terms of the BASE fare apply to the new ticket.