You should go back to the store and tell the owner to
fire the "clerk" who told you that! About the only things that are the same between the two are bullet diameter and primer size. As stated above, the .375H&H is a belted case, the UM is not but is bigger in diameter. Headspace on the smaller case would be controlled by the extractor, if at all. If you
were able to fire an H&H cartridge in the UM chamber, it could burst and send a nice jet of white hot powder gas back at you - after it cracked the stock, etc.

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