Rome Fiumicino to Termini Station

I would buy the ticket when you arrive; you have to walk past the various companies' ticket desks to get to the bus, anyway.

Is this to get another train, or are you staying in the neighborhood?

The Leonardo Express (train) is the quickest and easiest method of travel to Termini - 32 minutes and it's not affected by traffic. Last time I used it the fare was 14€ PP, but DON'T forget to validate your ticket or that trip will become considerably more expensive. The same caveat applies for using the Metro in Rome, Buses, or other forms of transit.

As you point out, the bus takes about twice as long as the Leonardo Express train but costs about a third the price. I almost always use the bus when I'm not in a rush, and imo the scenery along the way is more interesting. The savings to buy online is pretty small, so I just buy my ticket when I get there. I look to see which bus company has the next departing trip to Termini (they place a small sign on the counter) and the price is usually €5 for a one-way ticket.

I almost always use the bus when I'm not in a rush, and imo the
scenery along the way is more interesting. The savings to buy online
is pretty small, so I just buy my ticket when I get there.

We've done the same as Frank, Chris. The little longer time to get into the city hasn't bothered us, the coaches are air-conditioned and very comfortable, and we enjoy looking out the windows as well. The ticket booths are just outside of baggage at the airport - you can't miss the signs - and we just walk up, purchase them, and are directed to the stand we need to be at for the next pickup: easy-peasy and cheap. Just a guess but Terravision and SIT are probably the most commonly used.

Take the bus, it is only 5 euros and takes about 35-40 min
Leonardo Express is about 14 euros (non stop) takes about 30 minutes
Train is about 7 euros (I think only 2-3 stops,not a big deal) about 35 minutes
Taxi probably cost 50 euros.
What ever you decide, buy it there at the airport.
Your choice...

The FL1 train does not stop at Termini, but at other Rome stations like Trastevere, Ostiense, Tuscolana, and Tiburtina. 8 EUR for all of those stops.