The Kindle Fire HDX 8.9" display is better than the iPad Air's.
From DisplayMate's comparison here:
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.
All the Kindle Fires have laminated displays while the iPad Air uses cheaper, older technology that has an air gap between the cover glass and LCD. Air gaps can cause eye strain among other problems:
Like all previous iPads, the iPad Air still has an Air Gap between the display and the cover glass, which increases the Reflectance from the additional three optical surfaces. The iPad Air has a Reflectance of 6.5 percent, which is a significant improvement over previous iPads, but is still 30 percent higher than the Kindle Fire.
The Kindle Fire HDX 8.9" is also lighter (374g,) has a higher resolution (2560 x 1600,) higher DPI (339 DPI,) and it's cheaper ($379 w/ ads, $394 w/o ads.) The Kindle Fire HDX's UI and app selection are poor, but it doesn't matter. I've owned several tablets, including an iPad mini, and I've found the only task tablets excel at, is reading. For everything else, smartphones, laptops, and desktops are better. Smartphones are more portable, laptops and desktops are better for productivity.
It's good to see Amazon putting a lot of effort into improving display quality. I don't like shopping at Amazon.com, but the Kindle Fire HDX 8.9" is an impressive piece of hardware that I'll definitely consider buying.