What is the difference between doing just one set and several sets in one training session, if you train to the same level of post training soreness?
Don't use soreness to plan sets
The soreness part of your question is distracting you from the proper purpose of set selection. You don't choose sets to find a particular level of soreness, because after the first couple of workouts, soreness is not diagnostic of strength gains. What matters is your ability to recover and lift more in the subsequent workout.
For the first workout(s), yes, pay attention to soreness/stiffness, and don't over-do it. But after that, soreness will lessen (even disappear), and should not be the key factor in deciding how many sets to do in a long-term strength training program.
This is more-so the case for low rep (<6), high weight (90% 1RM) scenarios.
In your case, doing up to 12 reps until you feel sore, you may actually experience soreness during recovery after every workout, but that's a side-effect of the rep range that you've chosen and because you're doing as many reps until you feel sore.
One set vs multiple sets
If you just do one set, that may not reliably reliably make you stronger every workout. It may work for the absolute beginner (which is why you can listen to soreness in your first few workouts), but after a couple of weeks, you need to be doing something that reliably stimulates adaptation. Doing multiple sets (3-5) is needed to reliably stimulate strength gains so that you can lift more in the next workout.
Here are some quotes from Mark Rippetoe and Lon Kilgore on the subject of sets:
Doing one set at 8 to 12 RM will yield about 80% of the potential gains from training in the 8-to-12 rep range. This may be enough to achieve the fitness goal of a typical health club member, or it may be adequate for assistance exercises after the actual barbell workout. It is inadequate for athletes trying to improve strength and power. (emphasis mine)
The number of sets must produce the metabolic effect desired as an adaptation. [...] If one set is all the work that is necessary to force an adaptation, then the athlete has not been training either very long or correctly.
One set of an exercise is not capable of producing the stress that multiple sets can produce because stress is cumulative.
If you're happy with getting about 80% of the possible adaptation between each workout, one set may be fine for you.