will the free quick-start rules be getting a potential update to reflect those differences?
I don't know. But it's not that the rules changed, but that additions were made. Like buildings, formations, scenarios, and explanation of not just how terrain effects movement/combat, but how to place/represent the terrain in AS.
Nothing in the current quick-start rulebook is wrong/changed (Usual caveat that there might have been minor errata to AS:CE that we went ahead and applied, but I don't recall any off the top of my head.)
But the AS Box set rulebook does not have the terrain templates and standees that the Quick-Start rulebook had (since those were separate items in the box set and thefore not part of the rulebook). So if we just replaced the quick-start rulebook with the AS box set rulebook, the new quick-start rulebook wouldn't be "playable" like the current one is.
So it wouldn't be a simple just replace the current quick-start with the AS box set rulebook.