Kind of a "what's your favorite flavor ice cream" question, Fred. We all have different preferences. Some people like Vienna more than Budapest or vice versa. Some people even dislike one of them.
Me? I visited Prague, Vienna, and Budapest in one trip, connected by train. I loved Prague, thought Budpaest was interesting, and thought Vienna was...pleasant. I totally fell in love with Prague, a compact walking town with oodles of charm (but lately, even more oodles of tourists). Vienna was next and was just a big letdown compared to Prague - it felt more ordinary, not special. That's probably unfair to Vienna - it's just that I loved Prague so much, and Vienna is different. I'm also not a big fan of museums or classical music.
Budapest was much more interesting to me than Vienna, but I found it hard to love Budapest. It's much bigger and more spread out than Prague, harder to get one's orientation in. I found Budapest felt a bit more edgy and darker in a sense. (This was 2005 mind you, I guess things could have changed.) I kind of want to go back to Budapest again (visited in 2005) and probably will someday, but it's still not one of my favorite places, so it never seems to make it onto future itineraries. I did go back to Prague once, loved it again, would love to go back in the winter when less crowded.
Some people absolutely love Budapest. Some absolutely love Vienna. I guess some love both. I didn't fall in love with either - but found Budapest more interesting. Not sure if that helps you, but there it is!