While neither city could claim to be one of the more forward-moving European capitals at the moment, at least Budapest used to be that in the past...at the time of Austro-Hungarian empire, everyone who is anyone used to send their kids to school to Budapest as well as Vienna and Paris...The high society used to enjoy a nice life there and it also attracted many poets, writers etc.
The old town has some stunning architecture whilst the new town displays the tyipcally communist design (bleak, functional (if that!), uninspired) that we also often see in Serbia.
The people can be moody although they would probably be nicer to foreigners. The food is very spicy although personally I love GOULASH and also in their supermarkets you can buy the most amazing cold meats and cheeses - in good old times, my mum & dad used to cross the border just to buy those goods from hypermarkets.
Some of my Australian friends went recently and were really impressed with the quality of bars, restaurants and felt that it was very Western and getting quite trendy, while at the same time the city still offers its unique geo-socio-historic character (because I don't imagine you want pure Western stuff, or you could just stay at home?)
Bucharest - I haven't been but I know someone who has, I think it offers less Western-type pursuits but it is still ridiculously cheap even for Serbs - but not generally known for any amazing sightseeing or even nightlife - it's worth checking an on-line guide to see what you would actually do there. My mum told me an anegdote of some 20 years ago when her and my dad went on a trip to Bucharest with his work, and were quite excited about going shopping (knowing that it was so cheap), only to find there was NOTHING in the shops and even to get a sandwich for lunch was a real misssion.
In my humble opinion, Budapest might offer a better city break...
Or, if you really want to have fun, go to Belgrade! At my recent wedding we had 40 foreigners, mostly English but also Australians, Irish and American, and they absolutely fell in love with Serbia and some stayed a few days extra to travel round.