As for the difference in mere sharpness, which has been repeated here several times, I think comparing and measurebating on the "sharpness" alone is a bit pointless.
Yes, the AA filter of the Bayer sensor has a significant impact on the perceived sharpness but, there's more to the Foveon sensor than just sharpness. There's a different look between the two, and which one is 'better' is a matter of taste.
The thing in the Foveon look is not in the sharpness alone, which is usually a result of several different variables, the lens being one of them. They say god is in the details, and that's true for the Foveon look, too. Certain details just look better when rendered by the Foveon sensor and SPP software, even though there was not much difference in the sheer sharpness of similar images. On the other hand, when there's less light and the ISO level needs to go up, the Bayer sensor works better. Like said, horses for courses, and apples and oranges can both taste good.
I remember seeing a nice comparison made by shooting a piece of clothing, and that particular photo told the difference much better than a long and verbose post in this thread. I'll add a link to it later, if I can remember where exactly I saw it. Anyway, it was quite visible that the difference was not about sharpness alone.
I've got both an AA-less Bayer sensor camera (Ricoh GXR) and a Foveon sensor camera, albeit a one using the older, slighlty smaller resolution version, and I've seen pretty similar results. Whilst the difference in sharpness is slightly less obvious, and the older Foveon sensor has slightly less dynamic range and resoluiton, there is still a similar looking difference. But I don't think that measurebating between the two is particularly interesting.
Suffice to say that there is a slight difference, and which one is better is a matter of taste. There really is no need to rank these two different systems in any order, unless measurebating is something one prefers over taking actual photos. I for one would like to have it both ways, which is probably why I ended up having both. Then there's still the third option, (for now), the reshuffled Bayer sensor á la Fujifilm. Something for (almost) everyone.
BTW, I've also seen a comparison made with the same kind of Sony sensor with and without AA-filter, using the exact same lens. In that comparison, the 12mp AA-less Sony sensor of the GXR Mount made sharper/better looking images than the 16MP sensor of the Sony NEX 5n, both using the same Voigtländer lens. But here again, there was probably more to the end result than just the AA filter, or lack thereof.
Even though they are both Bayer type sensors made by Sony, there are more differences than just the AA-filter. The image processor after the actual sensor is one of the differences. Besides that, the lower megapixel sensor, as well as the GXR mount itself have been optimised for the classic rangefinder lenses, unlike the NEX system. It has clearly thinner IR filter as well, very much like in the Leica M8, for example, and the module uses a native M bayonet, and doesn't need any third party adapters between the lens and the mount.
All these details result to an optimised, and in this case, better looking result than that given by a seemingly superior (in megapixels) sensor. I think it was a good reminder that it's not just about megapixel count. There's much more to nice looking photos.
Besides, if we happen to get pleasing photos with any camera, lens and sensor combination, be it a Bayer or Foveon or X-Trans based, perhaps we should just embrace it, then go out and take some more great photos, instead of pixel-peeping the differences inside, and online.