I'd love it if somebody could add a bit on what the heck use it is.  My experience so far involves a physicist who always worked with A.  I was running physical experiments on the device he had derived a theory for.  He insisted the device would have "no magnetic field".  I whipped out the gaussmeter and measured the magnetic field.  He would not accept the readings, as he considered the B field an illusion.  Yet he was unable to direct me to an instrument that could measure vector potential, and could not convert A to B due to the pesky gauge factor problem.