I'm a passionate lover of both cities (returning to Barcelona in April; Florence in June) and a three-day weekend is a great way to be introduced to both cities.
Florence is a much more heavily touristed city than Barcelona, IMHO. May is a good time (before the stifling heat and crowds of June and July) but in Florence you'll still encounter fairly large crowds of every nationality in groups surrounding a pour soul holding an umbrella high in the air.
You can, I think, cram much more into three days in Florence (in terms of art history, European history, social and political history, etc) than in Barcelona. Florence is a much smaller city, geographically speaking. You'd be very hard pressed to walk from Miro park in the Montjuic area of Barcelona to Gaudi park and Sagrada Familia, but in Florence you can very easily walk from the main train station to anywhere and everywhere worth seeing (including the Piazzale Michaelangelo overlooking the city, which is a moderately strenuous 20 minute uphill walk out of the city center, but oh what a view after dark).
The flight connections are easier to Barcelona than to Florence. More often than not UK flights go to Pisa, which is about 40 minutes away (IIRC) from central Florence by train. Sometimes you can fly into Florence's very small airport, but that is still an expensive 20 minute taxi ride into the main city.
I also think hotels are a bit more expensive in Florence than in Barcelona, just because it is a smaller area and there are simply more tourists there.
Stil, despite desperately loving both cities, for a first trip to the continent I think I'd give a slight edge to Florence. In a smaller area you'll encounter more stunning, iconic sights than you would in a similar area of Barcelona, which makes a three-day trip feel more full IMHO. Just the stroll from the main train station to the piazza in front of the Duomo, then down the street to the Palazzo Vecchio and the Piazza della Signoria, then another short stroll over the Ponte Vecchio, and within 15 minutes you've seen the very major sights and can spend the rest of the weekend soaking it all in; whereas in Barcelona you'd likely be spending three days traveling between the major sights on the tourbus or on the metro.
One other note: IMHO the risk of petty theft in Barcelona is also a bit higher than in Florence. Don't get me wrong-both cities have issues with pickpockets and purse snatchers and the like, but I simply think that the risk in Barcelona is a little higher and you're forced to be a bit more vigilant.
I'd be extremely happy to make more detailed recommendations for either city depending on which you choose. In the interest of full disclosure I did live in Florence for a while and am still good friends with some native Florentines.
ETA: The connection from Pisa airport is a little longer than I thought. You can take a 10 Euro bus direct from the airport to Florence Santa Maria Novella (central train station) that takes 70 minutes; there are also train connections and bus/train connection combinations that can take 60-100 minutes, depending on timing.