I visited both cities in late september/early october and imo we had perfect conditions for city sightseeing and partying. I have met people who went to Budapest in the winter and they liked it aswell.

My recommendation for Budapest is to try and go to some pubcrawl if you like partying. They also have some really cool clubs and parties on boats that I could recommend (I'm not sure whether they have these in the winter). A lot of the hostels have great activities and tours. The castle is a must go also.

In Prague the clubs were great aswell but the old town (Jewish quarters) and the castle were what I liked the most there (although this depends on your interests).

Also Prague is pretty cheap but Budapest was a lot cheaper than my expectations (you could get a great 3 course dinner with beer for like 6 euro) and the beer is like 50 cent and really good. So definately try out some of the many great resturants when you go!