Sorrento and Amalfi in early february

I haven't been there in February, but check a guidebook or hotel booking site for availability. I am pretty sure life doesn't stop for the winter, but there will be shorter hours, unreliable ferry crossings to the islands, some places may be close for the winter but some places will be open.

It will be quiet but open. It is a viable town and lots of people actually live there. Pompeii on a sunny, cold winter day is a delight as there are few tourists. Amalfi Town is super quiet, but again, if it is sunny you can enjoy sitting outside and having lunch, walking to Atrani (see Rick Steves' book for directions) and seeing the view from the bus on the drive to-and-from. I would not count on ferry service to the islands, but it could happen.

We were in Amalfi in early March of this year, taking a bus the hour from Salerno on a daytrip. Temps were moderate, approaching 60 F on a sunny day. We sat outside at the Lido Azzurro restaurant on the water, which was its first day open for the season. Quiet, beautiful, perfect. And I echo what Laurel said about Pompeii. We visited there the next day, with similar weather, and had few tourists to contend with. Fantastic.

I was in Sorrento for a few days, second week of February, just after the snows in Rome, so it was unusually cold and wet, but still didn't feel like winter and I had some sunny days. Shops and most hotels were open. More locals in restaurants than tourists, the Circumvesuviana train was never crowded, nor were Pompeii and Herculano. Another benefit was really affordable hotel prices - nearly half what they are a couple months later.