The gauge is a measure of the thickness of the hinge metal.
source: Stanley 5 Knuckle Ball Bearing Hinges
This would affect the size of the knuckle as it wraps around the hinge pin.
size of knuckle = (hinge gauge x 2) + hinge pin diameter (+ a bit to allow for clearance)
Heavy weight gauges increase available bearing surface area. It will also determine the depth of the cut-out recess(es) in the door/door frame necessary for a flush installation.
The gauge also determines the type of 'hinge tweaker' tool you need.
Hinge Tweaker Heavy Weight Size for .180 Gauge Hinges
You need a more robust tool with a wider 'mouth' to 'tweak' .180 Gauge Hinges than .123 Gauge Hinges