We here in the western USA would need a lot more buffalo to have a valid way to compare the 2 cartridges. But out largest common game is the elk. Moose are nearly as rare for hunting opportunities as buffalo.
I absolutely love my 375. In fact I rate it as one of my 2 favorite cartridges I have ever used. So much in fact that I am on it's 2nd barrel due to shooting the rifling out of the throat of the first one. I have used it to kill a LOT of game.

The 9.3X62 started to get a following in the Rocky Mountains about 7 years ago and I can't say what caused it to grab the attentions of the hunters in Wyoming and Montana, but 10 years ago most American had not even heard of it. It's got a very good following now. I have re-barreled a bunch of 30-06s and 270 to 9.3X62, and made fully custom rifles in that caliber for hunters around here quite a few times also. I got to see the effects on elk about 10 times before I decided to make one for myself.

So far I have seen probably 15 elk killed with the 9.3X62 and also my own 9.3X74R. I load both of my rifles with the same bullets (286 grain) at effectively the same speeds. 2394 FPS average for the 9.3X74R and 2407 FPS for my 9.3X62.

On elk, there is NO difference at all in the effects of the 9.3s and my 375H&H that I can see.

But elk are running from around 475 pounds up to about 900 pounds. If I were to start killing several dozen buffalo of 1600 to 2200 pounds, or moose from 1200 to 1800 pounds, maybe my old love, the 375H&H, would show me some difference.
But on elk, deer, antelope and bear, there is NO difference at all.

I'll just have to take your Aussy's word for it when you tell me the 375 is superior, but shooting wild cattle and water buffalo, you'd would have a way to see a difference I. Here in the Rockys, a man would have to be VERY VERY VERY wealthy to be able to hunt enough moose and buffalo WITH BOTH CARTRIDGES to be able to have any valid opinion. To get such a background I would say someone would have to shoot about 15 moose and 15 buffalo WITH EACH CARTRIDGE with the same kind of bullets in each, before you could have enough personal history to make a judgement for each one.

I suspect that if an animal was large enough to "challenge" 36 cal or 37 cal 270, 286 300 or 320 grain bullets, the 375 may be more effective simply because weight for weight, the 375 is a bit faster.
But any honest evaluations would be forced to admit from over 100 years of use in Africa ans northern Europe, the 9.3X62 and 9.3X74R are both quite capable in the hands of men who can shoot.