Both are 7-inch tablets running Android operating system. The Nexus7 has the latest hardware/software, whereas the Kindlefire is a couple years out-of-date (yes, KindleFire is 6 months old, but it's tech is older). Amazon is rumored to be refreshing KindleFire soon.
The weight makes it much easier to hold. The higher resolution screen is much nicer to read. The battery life is slightly longer.
It's not just the device, but also the cloud, and the social. All your Kindle content is held within the cloud, as is your "Google Play" content. But Google has a lot more "social" features. When I turned on my device, its top "Recommended Apps" was "Orbot: Tor on Android". That's because in my Google social circles are people who downloaded this app and recommended it. It was a weird bit of mind-reading, as that was indeed top of my list of applications I wanted to download (after the Kindle app).
|Nexus 7||Kindle Fire|
|Android OS||v4.1 (Jelly Bean)||v2.3 (Gingerbread)|
|Weight||340 grams||413 grams|
|Storage||8 gigabyte||8 gigabyte|
|CPU type||ARM Tegra3||ARM OMAP 4430|
|CPU speed||1.2 GHz||1.0 GHz|
|RAM||1.0 gigabyte||0.512 gigabyte|
|Battery||4325 mAh||4400 mAh|
|Battery video||9 hours||7.5 hours|
|Battery e-book||10 hours||8 hours|
|Camera (front)||1.2 megapixel||no|
|Input: ambient light||no||no|
One possible reason to get the KindleFire instead is that it's more focused on eBook reading, and therefore possibly better as a gift to somebody who is computer illiterate and just wants to read books.