Here's a comparison, focusing on the 7 inch Nook HD (and not the 8.9 inch Nook HD+) vs. the Nexus 7...
The 7 inch Nook HD (discontinued on 6/25/13):
The Nexus 7 (2013):
Yes, you can use the Nook app, which is available on Google Play.
Partly because the Nook has has ceased production on June 26, 2013 due to poor sales, the Nexus 7 is a better all around tablet, not to mention that because it is coming from Google, it receives unrestricted OS updates.
There is not much difference but many people prefer the Nexus 7.
There are various types of Nexus devices, but the Nexus 7 is part of that same family.
The 2012 Nexus 7 came out on July 2012. The 2013 Nexus 7 came out on July 2013.
The 2012 Nexus 7 debuted on July 2012. The 2013 Nexus 7 debuted on July 2013 (August 2013 in Canada).
Yes, the Nexus 7 (2013) is Bluetooth Smart Ready certified. The 2012 Nexus 7 supports traditional Bluetooth.
The Nexus 7 is an Android Tablet made by Google. It has a screen size of 7 inches.
The Nexus S 4G is a CDMA phone and the Nexus S is a GSM Phone. This is the basic difference between these two phones. Most of the remaining configurations are same.
In the 2012 Nexus 7, it came with a single camera (in the front). The 2013 Nexus 7 has two cameras, one in the front and one in the back.
Nexus 7 tablet 2013 Has a front and rear Camera.
Yes, the Nexus 7 comes in LTE versions.