Although I could be swayed by a strong argument for Schönbrunn, for now I'm plumping for the underdog.
Dresden's Professor Brandes House is stylish and quirkily themeless. Primates are its focus, but the koala, armadillo, squirrel, etc. are nonetheless well presented. It's also the most successful application I've seen of the sloth-on-ropes approach, which stretches throughout the house and into an outdoor enclosure (a nice touch), drawing an otherwise disparate collection together.
Vienna's small rat house is cool, but I prefer the rat sewer in Dresden's innovative underground exhibit. And, whilst the aforementioned elephant shrew enclosure is wonderful, I wouldn't count the Wüstenhaus as part of the Tiergarten's collection.
Perhaps Schönbrunn's stature is a handicap here, but I think Dresden deserves the win for how favourably it compares to one of Europe's greats.
EDIT: I deliberately didn't mention Vienna's South American exhibit, but since
@SabineB has brought it up, I think it's fair to say Dresden's screwed. Nice zoo, though.