I think that the notion of "shock absorption" in a shoe can be misleading. That diagram above in charliefedererer's post shows misaligned ankles which can occur in any shoes that don't match up correctly to a player's feet, including sneak's designed for tennis. When I've found shoes that give me proper ankle alignment, I also get much better "shock absorption" from them, even if the soles of those shoes are especially thin.
Pounding on misaligned ankles will drive any of us straight to tendonitis town. I think that this is why it's important to pay attention to the descriptions of different shoes. Some are made for medium to wider feet, others for more narrow feet, and even for those with flatter or higher arches.
As far as shoe stability and resisting ankle rolls works for me, I describe that by comparing an F-1 race car with a high-riding SUV. Turn hard in one direction or the other (yes, laterally) and which one is more likely to roll? The one that's got a higher center of gravity - the SUV. No comparison. My more stable shoes have been the same in that the closer they keep my feet to the court, the more stability I enjoy with them, even if the heel counter isn't built like the Alamo.
Running shoes usually have a lot of "stuff" under the heel, which might be good for a runner who needs to have that cushy business under the heel to soften the repetitive landing when loping along. As a tennis player though, that only creates a higher platform that my heel can roll off of when I push sideways good and hard. The lower my foot is riding, the less leverage there is between my heel and the court to effectively "tip my ankle over".