There is a subgroup of watchmakers which assemble the Special-designated watches. I can't find the interview right now, but I read an interview with a watchmaker who was working to try to achieve this status.
As for parts - in Seiko watchmaking videos I've seen, for each part there is a tray to choose from, and I'd guess the Special-designated watchmakers may be slightly more selective among them.
For quartz-regulated watches (like Spring Drive), Seiko has said that Special-designated watches get the very best crystals that Seiko grows. I guess even in homegrown quartz, variability remains.
Some special-designated watches receive distinguishing features (other than a 'Special' dial) - like a distinct rotor with the GS Lion - calling back to the early Grand Seiko chronometers (3180/43999/62GS).
It's not the case where they series-produce movements, and the ones which test the highest get the Special badge. The Special watches are made by special watchmakers.