Auro 3D is channel based, with specific speaker locations. Atmos is an object based system, so it does not require a specific number of speakers. It maps the number of speakers and their location in each playback system, and scales accordingly.
Auro uses the conventional 5.1 configuration of L/C/R/Ls/Rs, adding Left Height, Center Height, Right Height, Left surround Height, Right surround Height and Ceiling Voice of God channel. With the subwoofer channel, that's 11.1.
While Dolby has not specified any home playback specs, we do have their cinema guidelines
. This should give some insight into what the home system requirements will be.
There is no simple answer as to how to wire for both systems. There are significant differences in speaker placement between Auro and Atmos. Auro has height speakers on the front and side walls, with no back surrounds. Cinema Atmos does not use front height speakers or side height surrounds, but it does allow for multiple back surrounds and additional side surrounds located between the existing surrounds and the screen. Atmos also uses two arrays of ceiling speakers not on the center line but located a bit less than 1/3 of the room width from the nearest side wall.
Auro 3D will be available for home theaters in a few months. I don't have any inside information as to when Atmos will be available.