It's the same. In Premiere Pro, one small part of the process is accelerated by the GPU (but not the part that takes the most time, so it doesn't make much difference).
Use whichever application you're most comfortable with.
Warp stabilizing depends on the clip more than on the app used. It's a good tool for some shots but it's not a cure all and the more movement in the shot, the more warping is required, the more the shot will deteriorate.
As far as I know Warp Stabilizer in PPro is identical to WS in AE on the back end. I usually use AE's Render Cue to render to production codecs because I know the controls and it seems to work a little more efficiently for some tasks but PPro has the another pipeline to render and uses AME to build the render. AE also can use AME but I'm not sure what would give you the best results.
Stabilizing a shot is usually the least of the treatments I put on my work. Color grading is the most important visual part of any of my projects after composition and blocking. I think you're worrying too much about this. Both should work just fine.
I'd mention that for certain types of shots Mark Chrisiansen states that the AE point tracker (as well as the Mocha AE planar tracker) may each produce better stabilization results than the Warp Stabilizer in CS6, especially if you want to have the image stabilized around a certain point or area, rather than leave those decisions to an automated process.