DisplayMate evaluated the devices on a wide array of fronts, including calibration, sharpness, brightness, and color gamut. They found that while the tablets packed about the same density of pixels, the Fire HDX and the Nexus 7 were both able to display 100% of the color gamut but the iPad Mini was only able to show 63%.
The Mini was also dinged for being more reflective and for having a lower maximum brightness and contrast. And finally, the iPad Mini is also less energy efficient, with the Nexus 7 having the most efficient screen of the three.
It's not clear whether the Fire HDX or the Nexus 7 came out a head (it looks like a tie to me), but it is clear that both Amazon and Google have improved the screens on their tablets in a very short while.
In fact, if you discount the first Kindle Fire as a rebadged BB Playbook then Amazon and Google bested Apple after only released a couple 7" devices each. That shows a remarkably sharp ability to learn and adapt, and it should be a concern for Apple.
These test results come only a couple weeks after the iPad Air was dethroned as the best large tablet screen, leading me to echo DisplayMate's comment: "Apple was once the leader in mobile displays, unfortunately it has fallen way behind in both Tablets and Smartphones. This should be a wakeup call."