Both are great tablets for sure. My wife has the 7 HDX and two guys at work have The Nexus 7's so I have spent a decent amount of time with both.
The only advantage I feel the HDX has are the special offers. Those alone make the HDX worth owning: I bought Madden 25 for the PS3 for $5, a kindle Paperwhite for $20, and a Keurig coffee machine for $20. All in the first month of owning one. The HDX basically pays for itself and then some just because of the money you save on the sales because of owning one.
Overall I like the Nexus 7 better though. I find it to be much more comfortable to hold than the HDX. Although the HDX runs just as fast and smooth as the Nexus, I really don't like the OS on the HDX, while I absolutely love stock Android. Lastly, I hate that the Google Play Store is not on the Kindle tablets, I hate it.
----------
It is almost impossible to dismiss the issue regarding the screen display on the 7" HDX that is not an issue on the 8.9". Not sure how prevalent it is but buyer beware I suppose if it is aplenty.
It was so obvious on the display model at BB that one cannot miss it if their unit does have it.
Maybe we got lucky with our HDX 7, but we have to look really hard to see the blue/purple around the edge of the screen. It's there but it is so small and faint that with normal use we never notice it all. My wife and I are very picky about things like this but it has been totally a non issue for us.