In response to some recent, and to my mind very unfair, Naples-bashing in this forum, here are some concrete positive reasons for visiting Naples. I've been there twice for a total of 13 days, both times in April (so I can't comment about summer heat and tourists).
1. Art. The Capodimonte Museum is wonderful. Just the Farnese Collection itself is full of gems, and then there's the interesting exhibit upstairs showing the development of painting in Naples, with a Caravaggio here and several Riberas there along the way. The current exhibition of Salvator Rosa was a wonderful discovery. The Archeological Museum is unique - worth the trip for anyone interested in antiquity, and an essential complement to Pompeii. The Capella Sansevero has sculpture that almost breathes -- astounding! The Caravaggios in the Pio Monte della Misericordia....I could go on and on.
2. Architecture. Everywhere you look there's a beautiful archway or interior courtyard or baroque facade. Sure, some buildings are run down, others have been lovingly restored -- but Naples is a city that LIVES with its history and architectural heritage. All those baroque churches, the cloister of Santa Chiara with its exquisite ceramic tiles, the Certosa San Martino (also worth the trip in itself).
3. History. 3000 years piled up layer upon layer. One reason (aside from pizza and gelato) that Naples is an interesting destination for families with children old enough to be interested in history. Looking into the excavation at the future Piazza Municipio subway station is fascinating!
4. Scenery. Is there anything else like it? The view of the Bay of Naples, with Vesuvius and the Sorrento peninsula in the background, is one of the great sights in the world.
5. Food. Where to start? The pizza marinara at Da Michele? Lemon anything? Pastiera (a ricotta-based cake that manages to be rich and ethereal at the same time)? Ah, spaghetti alle vongole (with those delectible baby clams, pezzogna (equally delectible local bream) poached in "acqua pazza".... I have to include a paean to Ciro a Santa Brigida, an institution of a restaurant where everyone from tourists off the street wanting a pizza to local dignitaries are treated with courtesy, warmth and professionalism, and the food is memorable!