From what I gather, the issue is really one of comparing apples and oranges. Sinn Tegimented steel has a hardness rating of around 1200 Vickers / 72 Rockwell (from the Sinn website). But the process achieves that hardness over a thin 20-30 micron layer on the surface of the metal.
Ice/liquid nitrogen-hardened Damasko steel cases are in the 800 Vickers / 64-62 Rockwell range for hardness. But this process treats the entire metal right through, as opposed to a super hard exterior layer.
Which one ultimately is better from a scratch resistance standpoint, I really don't know. Damasko claims that just hardening the layer leads to an "eggshell" effect, where the surface may not scratch per se, but enough outside pressure may still dent the metal because the underlying steel is not as hard as the surface. Damasko says that because all their metal is ice hardened, this is not a problem with their cases. I would expect though that in 99% of regular use cases, simple day-to-day scratch events are covered adequately by both methods.
One thing to note is that for Sinn, you have to look for the watches that offer tegimented steel cases. Not all watches in the Sinn lineup are case tegimented. For some models, it's only done for the bezel (for example).
I believe all Damasko watches use ice hardened steel. And the steel is used not only for the case and bezel but also the crown and any pushers.