The Nexus 7 is much better than the 7" Fire HD overall, they have the same screens but the N7 is much more powerful, has access to the Play Store, and runs a more capable and fluid os. The Fire HD 8.9" is more interesting but the SoC is far to weak to drive that resolution.
I don't think it's so clear cut.
Kindle Fire HD has a better looking screen with richer colors, deeper blacks, and less glare. I think some Tegra 3 bug is causing our Nexus 7 to look a bit washed out.
TI OMAP 4470 looks like it might be able to hold it's own against the Tegra 3, hope Anandtech can test that out soon. GPU looks like it might be more powerful and a review of the Archos 101 XS using that processor showed it looked competitive in a few benchmarks with the Nexus 7 (though I didn't see enough to really know for sure). So I'm not discounting the TI OMAP 4470 yet.
Kindle Fire HD starts at 16GB. No silly gimped 8GB model (with only 6GB of usable space) and no $50 for an 8GB upgrade. Google did this intentionally to push people harder to use it's streaming content, but it kinda sucks.
Better wifi support, I especially love supporting the 5GHz band. My Nexus 7 tablet is the only thing I have stuck at 2.4GHz (I have almost a dozen 2.4GHz wifi networks competing around me and I'm the only 5GHz network).
For hardware, I think the Kindle Fire HD wins pretty easily. The SoC battle looks too close to call so far. And possibly the one hardware win for the Nexus 7 is it's physical design, it has less width in portrait, so will be easier for more people to hold one handed with thumb and fingers on both edges of the device. And the Nexus 7 is lighter.
For software, I'd give the Nexus 7 a dominating win. And I'm an Amazon Prime subscriber, bought all my books and music from Amazon Prime and I stream all my online TV and music from Prime now (cancelled Netflix recently). But I love Jelly Bean, it has given my Galaxy Nexus new life and the Nexus 7 is definitely buttery smooth compared to other Android tablets I've had. So it would really suck to be stuck on Amazon's version of Android 4.0.3 for who knows how long (forever?). I never touched a Kindle Fire, but from what I see of the UI, I want to hurl (can a real launcher be run ontop of that crap if I rooted it?).
I still think the Nexus 7 wins, but I think Amazon presented us with some really tempting hardware. I might consider getting a Kindle Fire HD 8.9 once folks get that rooted and have a decent ROM ready for it.