The difference between the .375H&H and 9.3x74R is analogous with the difference between the .270 and the .280. To some people the difference is significant, to others it is not. I prefer the .375 but I have both. Both are great all-around cartridges.

Two years ago a buddy and I were dropped off in the wilderness of Alaska for a week of Caribou hunting. He took a .300WinMag and I took a .375H&H. Every time we got into a place where there was potential to be ambushed by a bear, my friend wanted me to go first. In such an environment I would regularly choose the .375H&H over the 9.3x74R but I still wouldn't have felt under gunned if I had taken a 9.3x74R instead.

NitroX makes a good point about the guns. If you want a lighter double then I don't think the .375H&H is going to be a choice unless you go with the flanged version. On paper, the 375 flanged loads are closer to 9.3x74R loads. But you will find many more double rifles to choose from in 9.3x74R than in 375H&H. Consider that the makers offering 375 doubles will build you a 9.3 but some makers of 9.3 doubles won't build a 375.