.357 Magnum loads will generate higher velocities, even in snubs. However, I don't live in a vacuum so just that aspect in isolation isn't a very practical question for me. Even if I could control it well, other factors include increased muzzle blast and flash. I suppose one could use a mid-range load, like the Speer SB-GDHP, but then the velocity advantage is reduced.
"Stopping power" is a very difficult concept to quantify. When someone mentions how powerful the 125gr .357 Magnum JHP is, I think of the LEO who was hit with that round in the heart, but was still able to overcome his (or her? I can't remember...) attacker and survive. I also remember reading a couple of stories of attackers who were stopped instantly with one or two rounds of Federal 125gr Nyclads, which some consider anemic for self defense. What I have seen is data showing that among the typical handgun service calibers, there isn't much difference between them in terms of how well they can stop an attacker. I probably wouldn't worry over whether I should use .357 Magnum over .38 Special in a snub. I'd just pick a good self defense load that I can shoot well and leave it at that.
Just my opinion.
FWIW, even when I had a 3" 65, it was kept loaded with .38 Special +P loads.