The Shoot-Out's author, Dr. Raymond Soneira, is the developer of DisplayMate which is recognized as a worldwide leader in video diagnostics. His wonderfully in-depth article tests, compares, and analyzes the displays on the current flagship tablets by Amazon, Apple and Google in copious detail.
Although the Amazon and Google slabs are higher resolution than the iPad Air (2560 x 1600 vs. 2048 x 1536 pixels), Amazon edges out even Google with a higher PPI rating (339 vs. 300 PPI) -- and both best the iPad's relatively weak 264 PPI.
All three of these Flagship Tablets have the most impressive mobile displays currently available. The Amazon Kindle Fire HDX 8.9 and Apple iPad Air are even more impressive on another performance level because they deliver top notch picture quality, absolute color accuracy, and accurate image contrast that is not only much better than any other Tablet, they are also much better than most HDTVs, laptops, and monitors. In fact, with some minor calibration tweaks they both would qualify as studio reference monitors.
Most impressive of all is the Kindle Fire HDX 8.9, which has leapfrogged into the best performing Tablet display that we have ever tested, significantly out-performing the iPad Air in Brightness, Screen Reflectance, and high ambient light contrast, plus a first place finish in the very challenging category of Absolute Color Accuracy.
While its 264 PPI "Retina" display was pioneering when the iPad 4 came out (in September 2012) Apple's decision to maintain the same pixel density in the iPad 5 (nee "Air") has cost it in terms of quality and allowed the competition to catch up.
Although it has yet to ship, it'll be interesting to see how the 2048 x 1536, 324 PPI panel in the upcoming iPad mini Retina compares in quality to the Kindle Fire HDX 7 and Google Nexus 7 which both rock 1920 x 1200, 323 PPI displays. See the CNET has a comparison table.