The new Contacts app still uses the unconventional "book" user interface introduced for Lion's Address Book, which laid out contact listings as though they were written in a physical, leather bound notebook, with groups accessible by clicking on an ornately detailed red bookmark.
The new Contacts in Mountain Lion tones that down slightly, removing the bookmark theatrics and putting Groups on the same page as listings (below top). It's still possible to set the view to omit Groups, or to simply show one contact at a time (below bottom).
A related fix that is greatly appreciated: when you enter a term in the search field and then click on a group (to narrow the search down), as long as the search is relevant (returns results within the group), it will show you matches within the selected group. Previously, every time you selected a group in Address Book, your search field was cleared and you'd have to start over.
There's a new Share Sheets button that lets you send (via its popup menu, above) one of your Contacts out as an Email, a Message, or to another nearby Mac using AirDrop. Just as with sending files from the Finder, AirDrop looks for other Macs on the same network with an AirDrop window open in the Finder, populating everyone it finds as potential targets for "AirDropping" the contact record.
Duplicates and Links
While there aren't any sophisticated new tools for managing duplicated data within your contacts, there's still the all-or-nothing "Look for Duplicates" feature that offers to blindly fix however many duplicates it thinks you have, without any way to review what it will be doing.
Because it only looks for matching names, this doesn't seem to be very useful, particularly if you can imagine a scenario where you have more than one contact with the same name, or if you realize you have duplicate data where the first name isn't going to match (say, separate cards for the same Bob and Robert with identical information).
However, there is a new feature in Contacts for linking multiple contact records associated with different accounts, such as a record in iCloud and another in Yahoo and another in a corporate Exchange address book that all relate to the same person. Once linked, all the records will appear to be the same card in Contacts, simplifying and unifying your listings and putting all the related information in one place. This doesn't change information on each account, and the card can be unlinked later.
Facebook support for Contacts (coming soon)
Apple's Facebook support for Mountain Lion (which won't ship at the launch of OS X, and isn't scheduled to ship until the fall) can additionally take all your Facebook "friends" and add them to your Contacts. Additionally, it will update their photos and contact information in concert with Facebook. Apple will also provide Facebook matches for your Contacts who aren't connected to your Facebook circle.
This doesn't happen automatically. Similar to Twitter support, you have to manually go to the accounts pane of System Preferences and log into Facebook just as you would for Twitter. You then have to provide explicit approval for Facebook to work with Contacts (or any another app that requests access to your Facebook account, again just like Twitter integration).
Once you do this, you'll be able to post Facebook status updates from Notification Center, get Facebook options on a variety of system-wide Share Sheets, and you'll have all your Facebook friends integrated into Contacts, with their URLs, addresses, phone numbers and other information they've publicly advertised about themselves within their Facebook account. Facebook also becomes a "Group" in Contacts, so you can sub-select it when searching. Additionally, you can jump to a Contact's Facebook profile or photos page from Contacts.
If you've already entered Facebook names for your Contacts, the new Facebook integration software for OS X will duplicate this field in your contacts. However, the information it puts in is badged with a "fb" logo so you know that's the entry that the software is maintaining automatically for you. Facebook data uses "links" to join users' Facebook cards with existing Contacts cards, but a new "Linked cards" entry (above) lets you click each account listing to see separate cards with only the information associated with each account.