Technically, I do not believe the two mean the same thing. Bore, means the hole in the barrel, Gauge means the size of that hole. We Americans are the ones always accused of "Not Using Proper Language". On this matter I think we are ahead of the Brits. It really doesn't bother me what anyone else calls it, for me though it's Gauge, see no reason to "Step Backwards". Might be noted the Germans as well as some other Europeans, at least they used to say, "Caliber 12".

Miller/TN
I Didn't Say Everything I Said, Yogi Berra