I put up a comparison between the Paradox (I used to own a pair, No. 18, when they still used socks to help keep the pads on) and my MD + dog pads a while back. While I was unable to do a direct comparison with the alpha pads, I think Dan has given us enough information about the changes brought by the new pads that some decent assumptions could be made about how the Paradox and MD + alpha pads would compare if one were going off my original comparison.
The most noticeable difference between the two headphones at the time was that the Paradox seemed to have more output in the 40-300Hz range or so. Yes, my Paradox sounded "thicker" and bassier than my MDs. I'd prefer something in-between the two, honestly. Since the alpha pads didn't change much in that area, those differences should still stand. I also thought the Paradox sounded a bit more refined overall, but that wouldn't surprise me given the additional cost and amount of time Luis reportedly puts into each Paradox (different business models, really). Other than that, they sounded surprisingly similar. How they compare with headphones like the LCD2/3 or SR-009, I have no idea.
I can't say everyone will experience the same, though I do feel like some exaggerate the differences between the two headphones. Dan tested my unit because I was surprised by what I heard in my original comparison, and it checked out (he was even kind enough to send me personal measurements of my pair, and they were very similar to his published measurements). I can only assumed the Paradox I had checked out as well, and Luis assured me that even the new pad mounting scheme on the Paradox (without socks) had very minor impacts on the sound of newer units at the time. I do want to note that I've read more reports on sound variability in the Mad Dogs units (not counting new revision) than I have heard about the Paradox.
Anyway, that's my thoughts on the matter, and your experiences might be totally different. I'm a fan of both products overall and don't think you can go wrong with either! Best thing to do is find a way to compare them yourself.