I think a lot would depend on the type and style of rifle. A few years ago William Evans in London had a very nice big game rifle in a 338 and I recall it was 338 Lapua Magnum. It handled like a proper big game rifle, and I have no doubt it would do just about the same job as the same rifle with .375 diameter hole down the middle. But in many African jurisdictions you need a min of 9.3mm diameter for any of the big five, so why choose a 338 Lapua over a 9.3 or 375. That William Evans would be a great rifle were longer shots are needed on say Brown Bear's, Moose etc in Europe or North America.

But many 338 Lapua Magnum are built as long range sniper rifles - think Accuracy International etc. They are built to be shot from a rested position - usually prone off a bipod - and are usually carried in rucksack type harness / drag bag. Not sure that I would want to carry one very far, especially not in the hot African bush. A few here in Scotland have 338 Lapua Mags - ideal for shooting gongs at 1,000m and the odd red deer way out on the open hill.

Personally for Africa, the 375 H&H would be the go to calibre for a very good all round rifle if you wanted something bigger than 7x57 / 30-06 class of rifle.