I can tell you from my historical research that by the time Williams Hand Tools was acquired by Snap On, they were nothing more than a paper company. Physical assets were liquidated during bankruptcies (yes, plural). They essentially bought the rights to the name and all the intellectual property. So there's nothing being made in any remaining Williams plant.
From what I've seen, some of the tools like the OP's sockets are indeed identical. When you think about manufacturing processes, it would actually cost more money to somehow differentiate what is otherwise an identical item cut from the same wire stick (or they're going to keep two different inventories of metal stock and two different finishing facilities to ultimately produce a virtually identical product? Not.). I believe the stories about differences in handling and finishes, alloy, hardness, etc etc yada yada, are just that--stories. Descriptions of distinctions in the invisible tool spirit realm. Yes, and if the Williams dealer want to keep his dealership, he better well regurgitate the party line.
Given the price difference, I know how hard it must be for Snap On buyers to accept the possibility they might be the same tools. Like the old cartoons where the character's head transforms into the lollipop sucker upon realizing what's really going on. There MUST BE something different about them. There's just gotta be something different. There just has to be.....
Maybe the Snap On marked tools are forged in a secret underground bunker and are polished with the finest silk cloths on the laps of Polynesian virgins. It's gotta be something...right?
).