All of the bridges! So many unique and cool bridges! I won't go into them here, but if you look up Budapest, you basically only know about bridges.
The food scene is also spectacular here, a great step up from Vienna! As a city, Budapest is what I would consider to be hipster, in the most raw form of the word. No pretensions, a little on the dirtier and older side, but still excellent.
Parliament, Buda Castle, Matthias Church
Really unique structures and places, worth noting and spending time in! The patterned design in the Matthias Church makes it stand out from the various others. Architecturally, these structures are an obvious mix of the East and the West, but less loud than Istanbul.
These are probably the coolest bars I have ever been in. The vibe is spectacular, the music is awesome, and even though it is touristy, it's still really cool!
Most of these European cities can claim a unique design sensibility, but Budapest really has a great collection and concentration of local designers featured in shops.
Head to mono art and design for a design map, a local walking map of design stores.
The ice cream is simply amazing here. And generally, Hungarian food is just local and absolutely delicious. They've taken the best food of various cultures and combined them into a great local cuisine.