I have been a fan of the .375H&H since my teens, thanks to its almost legendary reputation among gun writers and hunting writers. Used it in various rifles over the years for just about everything I hunted...deer, black bear, moose...and also groundhogs, coyotes, crows, you name it. Out to 300 yards it was perfect. I just knew I would go to Africa someday, and a .375 would be the gun I would take...and, of course, it would be in H&H variety.
But I eventually started to think more and more about a .416Rigby, and eventually got hold of a RSM chambered thusly. I used it for a few years, shot a couple animals and several varmints and predators with it. It became the apple of my eye, and morphed my African plans to include 2 rifles. I never shot an animal at much over 150 yards with it, but did a fair bit of work on targets up to 300 and felt quite confident about using it. Unfortunately, the dang thing split its stock at the wrist only a few weeks before I was scheduled to leave for Africa (yes, I finally went, and only about 40 years after first vowing to do so...) so I ended up taking only my .375 and never felt I was limited in any way. The H&H (in a Blaser R93) took Springbok at just under 300 yards and Cape buffalo at just under 20, with equal aplomb.
I recall reading an article by Ross Seyfried (only one author among many who wrote comparisons between the two) in which he strongly recommended the typical African hunter to go with a .375 if he only expected to shoot a single elephant or buffalo, and to leave the .416 to professionals and others waging extended battles with the big guys.
In terms of shootability and recoil, I didn't find much difference between the two; but almost all my .375's had been relatively lightweight compared to the massively overweight .416 in the RSM. All that weight must have mitigated recoil a great deal. I had owned only one .375 that probably approached the weight of the RSM (an older CZ) but didn't have them simultaneously so couldn't really compare them side by side.
Both cartridges were relatively easy shooters, much more comfortable in the recoil department and of course much more flat-shooting than the single .458WinMag I owned for a time. Both of the smaller cartridges could be considered very versatile and multi-purpose, while IMHO the .458 was too specialized and short-range for me to consider it an all-arounder.
I sold the .416Rigby immediately after replacing the stock, and never felt the urge to replace it. Now that I am gradually reducing my gun hoard, I am down to a single .375H&H and, with no more African hunts in my future, am unlikely to get another...despite the fact that I could easily get another Blaser barrel in .416Rem and be shooting for relatively little money. Unless you are still at the point where there are never enough guns in the house, I can't see the point of both a .375 and a .416 in the same safe; they're just too similar in capability. If you want something bigger and badder than a .375, the next logical stop is .458-caliber...and even then, its only real benefit is if you are chasing the biggest of big game. If you just like big bullets making big holes, the .45-70 is, in my mind, an infinitely more entertaining and easy-going cartridge.
I don't have any experience at all with the bigger .375's and .416's and other "supermagnums". I'll never need them and have no interest. I've never quite understood the fascination for more-more-MORE that leads every cartridge to be hotrodded to within an inch of its life, or beyond.
There are two other cartridges I've played with a bit which might be of interest in the context of this thread. The first is the .425 Express, essentially a .300WinMag necked up to take .423 bullets at about 2300-2400fps. The second is the .450-400, shooting the same bullet weight at, I believe, about 2000fps. Both were fun to shoot...the .450-400 is a total pussycat...and both are apparently very effective on game, as the numbers would seem to indicate. I just didn't keep them long enough to really get to know them; kinda regret that now.
As it stands, my gun collection is getting more streamlined with each passing year; I can easily see a day when a couple of Blaser stock/receiver units and a bunch of barrels will keep me happy. On that day, only one of the cartridges being discussed here will be represented: the .375H&H. There are no other close contenders.