best airlines to fly from Rome to Athens?

Agean air. Used it several times.

Not sure how you define "best". They are all safe. Pick the one with the departure time and price you want. Its only a 2 hour direct flight.

Safe means getting to the destination on time, good communication by employees, with no mechanical issues, etc.

I think they are all the same. We like EasyJet and Agean for the Greek Isle. Pick the schedule and price.

Safe means getting to the destination on time, good communication by employees, with no mechanical issues, etc.

To me, that might partially define "best". Safe is having a serviceable aircraft, adequately trained flight and cabin crew, and getting to your destination without damage to aircraft or people on board. As always YMMV.

There are 4 main airlines that operate direct flights from FCO to ATH:

ITA, the Italian national airline

Aegean, the Greek national airline

Sky Express, another (good) Greek airline

Ryanair, an Irish low-cost carrier

I've flown with all four airlines. From a safety perspective, I'd say all 4 are "best" since I've never been injured and I'm still alive.
As for being on time, all airlines experience flight delays, and most of the time it's not the airline's fault, so I don't see the point in trying to determine which one is the best in terms of "on time."

I should add that I don't like Ryanair, even though "safe and on time", it' the worst airline I've ever flown with. I even think "worst" isn't a strong enough word.

ITA (formerly Alitalia) went through a difficult period because they almost went bankrupt and experienced periods of many strikes and staff shortages. The company was partially rescued by Lufthansa.

Ultimately, I would say the best option would be to fly with Aegean or Sky Express.
I hesitate to say which is "best" because the cabin music during taxiing is better on Sky Express, but the cabin crew uniforms are better on Aegean.