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

In general you’d expect moving up within a company’s product line should yield improvements, especially if in addition to gauge you get other improvements in things like quality of the conductors, geometry, connectors, dielectric, etc.  So there are several variables at play here in addition to the gauge of the cable. 

As indicated by the Alan Shaw quote, the main factor affecting the performance of a speaker cable is the aggregate area of the conductors…to a point - as discussed by @raysmtb1.  IOW, if you have enough, then more will probably not change anything.   How big you need is affected by the current you are driving, based on the efficiency of your speakers and the output of your amplifiers.

However, the other cable materials and construction factors can also make a sonic difference.  As an example, my go-to choices here are both bi-wired cables, a stranded copper cable with two runs of 7awg to each speaker and the other being two 9 awg runs of a multi-strand, individually insulated, solid core copper cable.  I can tell the difference between them but I like them both and probably 4 times out of 5, I prefer the smaller, multi-strand, solid core cable.

if you are pushing a lot of watts, 15awg could be small’ish.  However, as others pointed out, you can go larger without breaking the bank.

Regardless of the quality of the copper, its treatments and the insulation used, and without considering capacitance and inductance, which is the speaker cable on the market with a larger copper section, therefore with less resistance for the same length?
(the final separation and possible reduction of a few cm to be able to enter the forks or bananas does not count)

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.