Florence in late February or early March

When traveling Europe, there is a week or two that the weather improves--going from Winter into Spring. We usually travel the last 3 days of March to get the winter airfares as they usually go up on 4/1 of every year. We usually have okay weather in April.
If you could possibly put your trip off 3-4 weeks, the chances of good weather will be much improved.
We were in Rome Easter Sunday of this year and it was pretty cold and we got rained on. This Spring has been extra slow in coming as we were very uncomfortable (cold) the last week in Budapest. Locals called it a "Russian Spring."

It's still winter weather, but Florence doesn't get as cold as Wichita, KS the same time of the year.

You are looking typically at high 50's during the day and low 40's at night. March tends to be very changeable and unpredictable, so anything can happen: rainy, sunny, warmer than usual, colder than usual, you name it.

The disadvantage of low season is iffier weather (as Roberto says, not "guaranteed bad" weather, but you must be prepared with sweater, jacket, and umbrella at all times). In places less popular with foreigners, English language tours can be curtailed or absent (probably not an issue in Florence).

The advantage are that airfare can be cheaper, and sights are less crowded. Note, however, that "high season" is longer than ever, as more people from all over the world are visiting places like Florence. The spillover from this means that times of year that were formerly dead are now somewhat busy, and times that were somewhat busy can be crowded. So, don't think you'll have Florence to yourselves in February or March, but it will definitely not be like July or September!

A corollary of this is that it is easier to get a hotel or apartment, with less advance planning. Some places definitely have "low season prices" and "high season prices" (usually indicated on their website). Others, however, have the same prices year-round; you have to check each place.

Another advantage of some places is that the cultural season is in full swing. Operas, symphonies and the like are often off during the summer and first half of September, but they're on a full schedule in February and March.

Two weeks before Easter in Rome was sunny and hot. Weather changed the day of the Rome marathon (April 2nd). Rain and much cooler temperatures. The following weekend in Florence it was sunny and hot. My hotel room for a quick weekend in Florence in February was much cheaper than the beginning of April but it was rainy and cooler.

We did Florence in late Feb/early March 2015, but it was only for 5 nights. Rented an apartment near the Santa Maria Novella train station for daytrips to Pisa, Siena, Fiesole, Venice, and walked everywhere. As stated above, you will not be alone, because there are plenty of students 'semestering' over there, and of course the parents come to visit. And no matter what time of year in Florence, you'll have the tour groups as well.

Being there for that length of time, you can definitely get into the Firenze vibe. But here's a thought. Why not split your 3 weeks in Italy by renting an apartment for 10 days in Florence, then 10 days in Rome? We just had an apartment in Rome in early March this year for 6 nights, and I am here to tell you that the vibe is completely different, and wonderful too. Unlike Florence, to get full measure of Rome you'll need to make use of the buses and Metro, but there is walkability too.

I found Rome different in that one can simply stumble upon ruins or a museum or a piazza that's been completely forgotten by the tour groups--can't 'sneak' up on something in Florence--too concentrated. Speaking of, I barely noticed tourists in Rome this time around because the city is more spread out and seems to absorb the visitors better. And you could easily day trip to Orvieto and Naples.

The two destinations would allow you to run the gamut art-wise, Renaissance to ancient. And although you will learn not to depend on good weather that time of year (it could be high 30's to low 60's, sun, wind, rain, or clear & serene), Rome could be a little more temperate than Florence.

But no matter what you do, it's our favorite time of year to travel to Italy, chiefly because we can do what we want, when we want and not have to worry about throngs of tourists. And if you're on a budget, the apartment is the way to go--you should be able to secure something nice for the two of you for no more $100-125/night in a great location in either Florence or Rome.

Think about it. Enjoy your planning!

We just wanted to thank you all for your thoughtful responses. They help so much! We are even taking Italian classes at our college now to have more fun with it.