Agree it is cheaper to book in advance, but......
When I came through Fiumicino a couple of weeks ago, within the space of an hour 6 flights had landed from USA, as well as one from Toronto, one from Santiago and one from Moscow, immigration for non-EU passport holders was knee deep in people despite about 10 or a dozen staff on passport control and the luggage reclamation area was full to bursting ( they have all the EU luggage too). If you got through all that in under an hour you would consider yourself fortunate . It is difficult to predict how long you need to exit the airport, never mind whether the flight will be on time.

In the context of the overall cost of a trip, I agree that the negligible saving on a non-refundable pre-booked ticket is seldom if ever worth the risk of a late flight or other transport delay.
It is difficult to predict how long you need to exit the airport, never mind whether the flight will be on time.
Yes I agree with all that ... we have taken a very long time to get through Immigration at FCO, which is a little strange, since leaving from there is fairly speedy.
If money is more important than time (and hopefully you're not having to arrive into Florence at night - it's a bit daunting) then book a Freccia from Rome Termini with good lead time, and just catch the first Leonardo from FCO that you can.

Thanks so much everyone for taking the time to help. All set to book the train on arrival. Two Australians really looking forward to our trip.

Well, you can catch the slow regionale train to Tiburtina, but there is no advantage in doing so
Well, there can be an advantage if that's the connection that's most convenient for you. No reason to outright dismiss it.
Some trains run by both Trenitalia and Italo bypass Termini and Tiburtina is their only stop in Rome on the way to Florence.
For any local train, including the Leonardo Express, don't forget to validate (time-stamp) your tickets before boarding.
Some trains run by both Trenitalia and Italo bypass Termini and Tiburtina is their only stop in Rome on the way to Florence.
Yes ... I think we established that well up-thread. And the savings can be more than trivial; if they need to wait only another hour or so to save a hundred euros or so, why not? Anyway, I think the OP was reasonably and happily informed by post #2.

Not to pick nits but the #2 on my screen is a thank you from the OP for a post that doesn't mention Italo trains or the Tiburtina station.
Discounted onward tickets on arrival day are always a gamble. If you don't plan to gamble, don't dismiss Tiburtina station as an option if it's the most convenient option.
By the end of post #3, the OP definitely was fully informed. The existence of Italo had been flagged, and assuming they are of average traveller intelligence, then the existence of Tiburtina as a possible third option to get them to Firenze SMN, should have been readily apparent from the timetables.
I think our job is done here, unless someone else feels the need to complicate a simple matter.