Kindle Fire has a better display and better graphics (or I hope so).
Nexus 7 uses the full google Android OS. Kindle Fire uses a gimped Amazon version that only supports Amazon apps and the Amazon App marketplace so there is a lot less OS wise. Although you can root your Kindle Fire to a better version of android.
I mean unless you really need that HD display, I still believe the Nexus 7 is the better machine just for the better processor. Sure the Kindle Fire HD has better graphics but comparing quad to dual core, the Nexus 7 wins.
Not true. The Kindle Fire supports pretty much all Android apps that would work on the device, you just don't have them show up on the Amazon App Store. Opera, for example, is just a simple download away. Sure, its still an android device, and hence its shitty with all the market fragmentation going on. But as far as android devices go, the Fire HD is probably the best you can get.
Also, a dual-core vs quad-core is hardly a difference performance wise. Better graphics and a better display are, by far, a better deal.