You can find plenty of forecasts on the web for the weather in Florence, but I have been searching for some information on the general climate conditions from season to season. While it seems a bit dated and the page has some broken links, this is about the best information I have found so far:

"To begin with winter (Early December-Early March), Florence alternates between crisp clear and cold wet days; daytime temperatures are generally above freezing, while it can dip below freezing at night, especially if the sky is clear. When a storm comes through it is generally a few degrees warmer, getting up to 6-7 °C (about 40 F). Snow is rare; the last time it stuck for more than a day in town was 1985, when the Arno also froze, for the first time since 1929. Outside of Florence temperatures are generally 2-3 °C cooler, and what comes down as rain in town often comes down as snow in the hills (this makes clear winter days after storms quite beautiful). So, if you will be here in winter, you will want to bring warm clothes, rain gear, good all-weather shoes, and an umbrella."

There is more here