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

You have to remember that not only speakers but to a lesser extent cables exhibit impedance, and together they form a impedance circuit. 

 To lessen that effect use thicker/lower gauge wire.

From Roger Russell (former McIntosh Director of Acoustic Research)

Wire Table

This is hard to beat.

Wire Gauge: 16 AWG
Capacitance: 35pF/m
Inductance: 0.52 µH/m

I needed some slightly longer cables as a new amp wouldn't fit in my normal amp rack space and needed to be a little further away from the speakers. I saw the new version of Kiimber 8PR cables now using the "varistrand" design (varied thickness little solid core wires in each strand), liked the look and paid less than 500 bucks for a terminated 13.5 foot pair. They look great and sound great...10 AWG to each speaker post. 

Depending on the amplifier and speakers, a larger speaker cable (lower gauge number) can improve the damping factor of the amplifier, which improves grip on the speakers. As to why/how a cable can effect sound quality, the construction of the cable, type of insulation, and whether shielded/unshielded can also impart audible changes. It's system-dependent so YMMV.