I think they are fairly different on paper. I just did a shoot out between my new SD-1 and a TS Mini. I was a skooch shocked at how similar they sounded. I figured the big difference would be in the feel of the pedal due to the asymmetrical/symmetrical clipping thing, but they really didn't feel that much different either.Just wanted to jump on this a little.
The SD1 clips asymmetrically to the Tubescreamer. They sound different and the more you turn the gain up the more you hear it. They can also sound similar (level up, gain down) The SD1 also cuts less bass and adds less mids. Pretty different.
There is also quite a lot of gain in both pedals - not heavy distortion, but smooth gain for lead playing. Also, a decent amount of level available.
SD1s have hardly changed, so try it and either it'll be the right pedal or the wrong one. Don't chase down different variants if the stock one isn't doing it for you.
Also try the MXR Badass OD. Similar thing. Maybe better as an OD. Less compressed.
The SD-1 has more volume and gain fer sure. Plus, the SD-1 almost rolled off as much bass as the TS. Not as much, but it was still quite noticeable. As far as mids, they seemed about the same.