For me the user interface was always snappier and felt just all around better. Now with webkit they have a faster render engine and feel like a lighter and faster version of Chrome (plus you can install all addons from Chrome with this extension: https://addons.opera.com/de/extensions/details/download-chro... )
Mouse gestures. I've tried gestures extensions in every other browser and found them all flawed or limited in some way.
Unfortunately other than that the new Opera is just a skin on Chromium. In the switch they've ruined password manager, fast DOM-preserving back navigation, and tons of others goodies.
I haven't used the new one yet, but the old one had a superior tab interface to everything out there, which they called MDI (Multiple Document Interface). For serial tab abusers like myself, it was a noticeable difference that always had me scratching my head why the other browsers didn't copy the idea.
Beyond tab management (which was exceptional), the side panel was and is my favorite UI widget hands-down. In the same vein as hot-corners, being able to 'toss' the pointer into the edge of the screen to open mail, notes, etc. felt just plain better than other methods.