Every app can use the notification system. They only can’t use push notifications if they are not in the App Store. Not terribly relevant for the desktop.
It probably opens and lists people on the left with their details on the right. It looks like a book. Can you group people? It's certainly not obvious that such functionality (which is useful!) exists.
Now click the red bookmark thing on top. Not very intuitive. It looks like a bookmark, not something that switches to a different view. Anyways, viola! The grouping interface! But now I can't see contact details. I need to "flip the page" back to see the details. This is a huge step backwards and is so inefficient that I completely stopped using groups when I moved to Lion.
Now consider Mountain lion: http://6.mshcdn.com/wp-content/gallery/apple-os-x-mountain-l...
Hey! It's the three column view from the old address book! Now I can see my groups, contacts, and details, just like before. Not as skeuomorph though. Now it's just an app that is skinned to look like a book. It doesn't function like a book. There are no pages to flip. It still has the dimensions of a book I guess, but that only makes that left page looks mighty constrained.
UI is about displaying and working with information, and this skeuomorphism is arguably not a very good way of doing either. I can lodge similar complaints about iCal. For instance, why are months always on separate pages? When I'm near the end of a month, I have to keep switching back and forth (with a slow unneeded animation on top). It's a computer, surely we can come up with an interface that discards this old page limitation of paper calendars! Not if you are thinking with skeuomorphism though.
If it’s only looks it doesn’t matter and is only about taste.
Further, looks are about much more than just taste. Designs affect how easy it is to understand the information being displayed, or to understand how to use an application. Skeuomorphic design is typically used to try to improve the latter: help people to understand how to use the app. It mimics a physical object because people already know how that object works. This benefit disappears if your skeuomorphic design doesn't actually work like the object it mimics though.
Skeumorphism that doesn’t affect functionality is purely about taste. Anything beyond that is bad. End of the story.
I would also argue that a lot of their designs look downright ugly to be honest.
The bookmark thing is a bookmark, and it works like one. It saves your place on a page in a book and you use it to go back to one. The little familiar groups icon on the tab tells you where it goes.
I'm not going to argue that some of the functionality of Address Book has been reduced or gone, but it has to do with a decision that Apple made about what they wanted, not directly tied to the fact that the interface is skeuomorphic now.