Will increasing speaker cable AWG make a difference in sound quality?

With respect to 'not much amps flowing'. If your woofer eats a 200W impuls at 4 Ohm that (read up on Ohm's law) makes it 7 amps. Some speakers dip well below 4 ohm, and some peak power burst can be higher than 200 W. That is the reason why sometimes amps are needed that can deliver lots of amps (pun intended). Not sure how big the differences are in the esotheric heights of $10+/ft price levels, but back when, even the least skilled listener could accurately blind test the 'lamp cord' vs a 4mm2 multistrand even on our  "LoFi" (Audiogon scale) systems. Interestingly, the by far best speaker cable we ever heard back then (and includes cables we would never spend our hard earned money on, but did A/B comparisons in listening rooms) was a coax cable used for cable TV distribution in large complexes (at 50 cents a foot from our wholesaler). Still have a section (40 years old) and eager to dig it out and power my Nautilus 804 with it, see what it does.