I have been to both Budapest and Barcelona. They are both great short-break destinations, but are very different!
We went to Budapest a few years ago now, around the middle of April, and it was beautiful. The architecture is stunning, especially Fisherman's Bastion overlooking the Hunagarian Parliament and the Danube. There are also different things to do like the Childrens Railway, and the baths.
Food and drink wise - I remember getting huge portions for not much money!
Barcelona, very different again - we too stayed just off Las Ramblas, and went to the Olympic village, and just wandered around. We went August time from memory, so it was pretty hot. While I enjoyed Barcelona, it's actually not my favourite Spanish city - I fell in love with Seville.
Both Barcelona and Budapest are great destinations, but I would imagine that if you were to go in February Budapest would be pretty cold!
I am sure you'll have a great time no matter where you go...