My first visit to Italy was 10 years ago, late November until a few days before Christmas. Seeing and taking part in how another culture experiences this major celebration is a lot of fun. The sights and towns can be more charming without huge crowds of tourists. I feel you get a better chance to interact with local people.
Neither the canals in Venice nor Lake Como were frozen. We didn't run into any snow until we went into the Dolomites, which was lovely. If you like to ski this is a great area to visit. And you have the advantage, on a clear day, of better visibility than summer so the long range views are great. So go ahead and go to the top of the Milan Duomo for a great view. You may get a rainy day or two, when all of a sudden umbrella vendors are everywhere.
From Milan south to Rome the colors are not brilliant as other times of the year, and dawn arrives late and dusk arrives early, but this shouldn't stop you.
Like others have mentioned, the lack of crowds makes travel easier and cheaper. Hotels and many attractions are less expensive too. Some things are closed or close early, but if you do your research well (read, read, read the RS books!) you'll have a great time.
However, if you are particularly sensitive to cold or really want to be in big crowds, winter would not be the time to go.