The joke and short answer is that S is for people who prefer to play games without maps, while G is for people who want maps, minis and all that jazz.
The slightly more involved but still not comprehensive answer (its a 356 page pdf after all) is the maps thing, plus not having to do math when building your mech, the inclusion of long-term resources to manage between fights, and damage/hp values being much better balanced. Those are the most important and most observable differences.