I'm seeing flights for ITA and Ryanair, but vastly different prices. What is the most reliable?
Thank you.

they are both top tier airlines and reliable.

Ryain Air is far larger, with in excess of 2500 flights a day all across Europe. In terms of passengers carried, RyanAir is the 3rd largest airline in the world - larger than Southwest, and many times larger than ITA

More important than reliable, it may make sense to look at flight frequency. IF there is a hiccup on your flight, when is the next flight from Rome to Bari that you could take? That might also weigh in my decision making.

Good point re: frequency - it looks as though neither has that many flights a day to/from Bari. Am I correct that both fly out of FCO?

Both fly from FCO, and it looks like both have 3 or 4 flights a day which is plenty of "what if" options. Some routes only have one flight a day, or only a few a week. Those are routes I get worried about.
RyanAir may have lower base fares but be sure to factor in thnigs like baggage (checked, carry on), seat selection, etc. My guess is apples to apples they are probably the same price in the end.

Since you sound like a newbie with RyanAir, you should know that it is essential to follow all the instructions on their website precisely, and in advance. For instance, checking luggage at the gate will cost more than on-line. Yet you can also trust them to keep to their schedule as closely as possible. Their financial structure is such that they can't profit from a plane sitting on the ground. They have mastered the quick turn-around.. But never fear; they have become one of Europe's biggest airlines in part through efficiency. You will get what you pay for.

Flown it and trained it. The train is more interesting as the views are better, the security less intrusive, the loos are better and there is more space. I'd take the train. (also no Boeing risk involved) ;-)

I'd say both ITA Airways and Ryanair have their perks. ITA usually offers a more comprehensive service, which might explain the higher price tag. They're known for reliability and comfort, especially if you're not a fan of super strict baggage policies. On the flip side, Ryanair is the go-to for budget travel. They can be quite reliable too but just remember, the low cost also means a bit less legroom and fewer frills. If you're traveling light and looking to save some euros, Ryanair could be your match.

Thank you all for your helpful comments. We had hoped to take the train, but our flight from the eastern U.S. arrives in Rome at 9:05 in the morning, so we wouldn't be able to make the 10:00 train and didn't want to kill the whole day with the transportation, so it looks like a flight

We flew Rome to Bari on Alitalia which is now named ITA. No problems at all. We have also flown RyanAir without any problems. Choose by schedule and price as both are reliable. I prefer flying to taking a train after a long day of flying from the U.S.

The train sounds lovely, but the 1:00 (approx) high-speed train leaves only from Roma Centrale - I don't see a connection directly from FCO. Am I missing something?
The flight leaves at about 1:00 p.m. also from FCO
Please advise if I am missing something.
Thank you.

I was assuming you were starting from Rome (city). From your latest question it seems like you are flying into FCO, and are seeking how to get on to Bari, your final destination. Buying a connecting ticket is fraught with potential issues, espeically "what-if" your incoming flight is delayed. If it's purchased as a separate ticket on ITA, you may, (or may not) be able to check your bag from your origin to your destination.

There are fast trains "Leonardo Express" from FCO to Roma Temini that run every 20min or so all day long, and take 32min. You do not need to book one ticket from FCO to Bari, but you will want to buffer enough time to get through immigration (this tends to move quickly and isn't a rate limiter), baggage claim (can be fast, or can take a long time), and customs. (no delay there as you just walk out the green lane unless you have something to declare. As with a connecting flight, the biggest unknown is when your incoming flight actually arrives.

some more info from you would help us give you better answers. . Starting point/ending point, airlines you've already booked, checked luggage or only carry on, etc...

Is your real question something like “flying into FCO & trying to figure out the best way to immediately proceed onwards to Bari”?

that’s a very different question than which airline is more reliable.