Personally, I think the answer to the question, "When should I go to Italy?" is "Whenever you can!" But life often conspires to make our choices for us. That said, I've noticed that travelers often avoid Italy in the off-season, and I'd like to put in a plug for visiting Italy when it's cold and grey. On purpose, even.
Sure there are disadvantages—there are less daylight hours to explore where you are, and you'll need to make sure that weather dependent locales will still offer enough services for your stay (i.e. we went to Procida at the start of November and most of the restaurants were closed for the season). Plus, you won't be able to pack as lightly with the cold weather gear that you'll inevitably need. But there are many benefits, such as:
1) It's cheaper. That is just inarguable. Flights are cheaper and accommodations are cheaper, and also easier to find since they don't book up as quickly.
2) Towns are lit up in a way that you usually won’t see if you google or use a guidebook. So places will look different than how you anticipated—lights of various types strung across streets, treats that are seasonal rather than ubiquitous in shop windows, presepe (nativity scenes) tucked into corners. Places will just look different than expected, and that will make the experience feel far more personal and far more magical.
3) Yes, winter can be cold in Italy. Very cold. But I have found that it is far more comfortable to be walking through Rome bundled and under an umbrella then trying to walk in a way that keeps your legs from sticking together and wondering how to stumble into a restaurant's misters just to get a moment's relief from the unrelenting heat. Making yourself warmer is easy with the right layers, but you can’t make yourself cooler, past a socially acceptable point.
4) You'll be able to witness customs and traditions that a summer traveler will never experience—like La Befana, the Christmas witch.
5) This seems like a minor issue while you are trip planning with such fervor that you have papercuts on your fingers from dog-earing pages, but it's huge when you hit the ground—breathing room. Having a clear view across famed piazzas and not having to fight for space to see into the floor of the Coliseum, those advantages cannot be overemphasized. Sure, you may have less daylight, but you'll waste less of it in tiresome lines, jockeying for position with a busload of tourists.
6) The problem with August, the prime tourist season, is Ferragosto. That little "holiday" that no one tells tourists about. It's unlikely to affect you if you are visiting big cities, but in smaller cities and towns, shops and restaurants and bars often take a week off in August.
7) Though there are, of course, wonderful festivals every month of the year, I happen to partial to those in the off and shoulder season. Probably because I'd rather celebrate truffles or freshly-pressed olive oil than the generic summer food festivals like ones celebrating pizza or foods in a giant pan (no joke). Off-season you'll find festivals celebrating salami, black celery, cinghiale, and chestnuts. These will give you a flavor of authentic Italy that you won’t get from waiting for hours to get into the Uffizi. (see bottom of post for videos from one of my favorite Italian festivals)
8) Rather than the usual tourist activities of walking from site to site or relaxing on a terrace with magnificent vistas and a freshly opened bottle of wine (okay, that does sound pretty good), off-season visitors can experience activities that tourists often miss. Skiing comes immediately to mind, though if you've read Il Bel Centro, you know I feel a little ambivalent about that one. Other activities that are the sole domain of the off-season traveler: tasting freshly pressed olive oil, Christmas shopping, ice skating on picturesque lakes, and drinking bombardino (a specialty of Italy in winter, reminiscent of eggnog).
9) Places that are too hot to visit in the summer become mild and appealing. Not all of Italy is too hot in summer and too cold in winter—the Dolomiti are fabulous in summer for hiking that is unparalleled in natural beauty, and the south of Italy softens and becomes comfortable in winter.
10) Italy is just more relaxed. Shopkeepers are less weary of tourists, restauranteurs are happy to see you, even the vespas seem more mellow. You'll get a window into what Italy is like when nobody is looking.