Genetically speaking, they're the exact same dog because Swiss Shepherds are direct descendants of white-coated German Shepherds, which were imported to Switzerland from the US, Canada, and England. However, as the white-coated GSD fell out of favor in Germany due to ignorance and bias, it was countries like the US, England, and Canada, where the dogs remained popular, that the naturally-occurring white coat in the GSD was saved from extinction. Meanwhile, the Germans were culling their puppies born with white coats, as well as breeding lines which carried the white coat gene, from their breeding programs. The German regulatory agency, the SV, eventually kicked the white-coated GSDs out of their registry, and thus kennel clubs in various countries started establishing these white-coated GSDs as a separate breed, even though they are still German Shepherds who just happen to have white coats. These breeds generally became known as, the White Shepherd.
White German Shepherds were imported to Switzerland in the early 1970s. The American White German Shepherd male “Lobo”, whelped on 5th of March 1966, is considered the progenitor of the breed in Switzerland, the White Swiss Shepherd. The descendants of this male registered with the Swiss Stud Book (LOS) and other White Shepherd dogs imported from the USA and Canada, gradually multiplied. There now exists a large number of purebreds over several generations of White Shepherd dogs throughout Europe. These dogs have been registered as a new breed in the appendix of the Swiss Stud Book (LOS) since June 1991. On July 4, 2011 the White Swiss Shepherd was formally accepted as a distinct breed by the fédération cynologique internationale (FCI or World Canine Organization), after having had provisional breed status since November 26, 2002. The country of origin for the breed is Switzerland.
Just like all breeds within kennel clubs, the White Swiss Shepherd has its own breed standards which it has developed. And, due to selectively breeding certain dogs within any breed, certain temperaments and slight changes to appearances happen over time. My white GSD has a very standard GSD face, which I love. I find that I personally strongly dislike the majority of US white GSDs overly slender faces, but I do see some on rare occasions that, like mine, have maintained the original GSD face.
There is actually a lot of misinformation and ignorance about white GSDs, especially within Germany, as well as within US working line breeders, and sports communities (such as IGP) where working line GSDs compete. Many Germans have never even heard of a white GSD, and I've even had Germans argue with me that they don't exist, which is ridiculous. Many of those in the states that breed and/or compete with working line GSDs mistakenly believe that white GSDs aren't suitable for protection work because they're soft dogs due to their white coat. This is ridiculous; there was a time when white GSDs used to trial alongside their black and tan, black, sable, red and black, etc. coated cousins and no one thought anything about it. White GSDs aren't softer dogs due to their coat color...they're usually softer dogs due to being bred specifically to preserve their coat color rather than specifically for their temperament. There ARE still some white GSDs that compete in IGP sports, but they're very rare due to so much bias against them within the sport world. If you look up "White German Shepherd Armada" on YouTube, there are a few videos of a Russian woman doing IGP with a White Shepherd. White GSDs also used to be on police forces here in the state.
Anyway, I hope this answers your question and helps explain some of the history behind the separate "breeds" of white shepherd.