Truth in lending, from a typical North American game animal's perspective, there is no meaningful difference between a group of cartridges such as the 7x57, 7x64, 6.5X57, 8X57, .270, 280, AND 30-06. However, the rifles in which many of these European creations are chambered can be very special indeed. The rimmed versions such as the 7x57R, 7x65R, and 6.5x57R are often housed in some of the most beautiful single shots and combination guns ever created - perfectly balanced and useful firearms that make pre-64 Model 70 feel like a pipe in a 2x4. This Bradshaw 7x65R is an example.
When one moves up into the realm of the 9.3, whether the 62mm in a bolt action or the 74R in a host of wonderful double rifles, they occupy a relatively low recoiling hard hitting niche not really filled by a corresponding caliber except perhaps the venerable Whelen. The 62 and 74R are something like the .338 Win Mag without all the drama. And of course you would have a hard time finding a 35 or .338 in a configuration like this pre-war German double.
The British, of course, pretty much defined the general purpose and dangerous game calibers for Africa, so I would assume your question didn't really refer to something like the .375.
My Rigby Highland Stalker in .275 (an Anglicized name for the 7x57) essentially does nothing that a Remington 700 won't do in .280. But that Rigby represents a level of tradition and craftsmanship somewhat missing in the usual Remington.
It is the special quality of so many of those firearms that is the attraction for me.