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

If the cable is Duelund, then the quality remains the same, but the sound signature changes.

https://forum.audiogon.com/discussions/duelund-speaker-cable

Duelund compared

I prefer the 16 over the 12.  Others don't.

I do wonder, however, whether some cable manufacturers deliberately (and unnecessarily) make their cables super thick to nudge the user's perception of them to one of power and muscularity in the sound.  

Good question. Good answers. Long story short, decreasing resistance in speaker cable with fatter wire can improve speaker performance… how much it might help is a nuanced consideration that depends on how much current reaches your speakers now and how much capacitance and induction your current cables hold. A couple previous answers work well to bring peace of mind . Well reviewed studio quality cable brands like Canare and Mogami are safe cost-effective universal solutions. My Wharfedale Heritage Lintons love Mogami 3103, a quad arranged cable with two 12 AWG OFC wires and two inert strands to keep the spacing even in the cable.

Distilled answer:

Maybe.  Maybe you can hear a difference, maybe not.

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. 

generally yes. Large wire size is required for the high current that flows for small fractions of a second during dynamic changes or big bass notes. a mellow song will sound the same with both but, just like in power cables, there are dynamic times when higher current flows, even if just for a picosecond.  The simplistic calculations like others are doing in this thread are wrong.  It is the google level of understanding physics.  I have a physics degree from UofC.