You can only go to one for the day. Which location would you rather visit, the Amalfi Coast or Naples/ Capri?

Where are you staying? Time of year?

And where on the Amalfi Coast? There are many towns on the AC.

Yes, it depends to some extent on the answers to the earlier questions but if you can afford to hire a private driver, then do the coast. If using public transportation, then probably Capri.

I will be in Rome during early October and will be using public transportation. I can take a train to Naples and then boat to Capri for pretty cheap. I'm not sure about the logistics of Amalfi.

The only thing I would consider is Pompeii - IMHO way more interesting than either Capri or the Amalfi coast.

That would be such a long day of travel for minimal payoff imo. I am going to Rome in October, and I've been reading up on places closer in for day trips, such as Gaeta, Terracina, Castelli Romani, the island of Ponza.

At that time of year, there is a risk when going to Capri that the weather could turn and the ferries are cancelled.

I agree with nytraveler and would add the archeology museum in Naples which is where all the artifacts found at Pompeii are now housed/displayed. But if rain is predicted for that day, I wouldn't leave Rome.

Yorkshire, Ponza is lovely once you get there, but I wouldn't consider it for a day trip out of Rome, esp. in October when ferry schedules are reduced and the same weather issues apply.

I replied earlier, Capri, if by public transportation BUT if you are based in Rome, Capri as a daytrip would be a bit much IMO.