Hi,
By my reckoning, the .375 H&H is one of the most versatile cartridges on the planet. Likely, it passes the .30-06, which enjoys so much because of its longevity, and willingness of companies to invest in so many loads due to its popularity.
As you get heavier, there is a tradeoff between exterior ballistics, terminal ballistics, and recoil. The .375 H&H seems to have retained all of its exterior ballistics with manageable recoil. By the time you get up to 300 grain loads, the .375 is starting to sacrifice its exterior ballistics.
The .375H&H is also much cheaper than the .416's.
As you get heavier, costs go up. Above the .375 H&H, the next cheapest and more powerful load is the .458 win (by my reckoning). It seems to have loads for from 300 gr (Federal dropped their 300 gr load, but you can handload) up to and above 500 gr bullets. At the 300 gr end, it will launch the bullets in excess of 2700 ft/sec (like the .416). However, the poor BC of .458 caliber bullets in 300 grains gets in its licks, and the bullets don't retain as much energy as the .416 downrange (perhaps some selection for better BC's would help: Barnes Originals come in 300 or 350 gr, and are semi-spitzers); Federal used to load 300 or 350 gr loads, but they dropped it. In 400 gr loads, the .458 matches the .416's (according to Federal's online ballistics charts; they load a 400 gr trophy bonded bear claw -- which is not [yet] available for handloading). Then you have the option of 500 gr and heavier in .458. But, the .458 has sacrificed its trajectory compared to the .375. The .416 seems to sit in the middle of the .375H&H and the .458. Not quite long range, and not quite 500 grain hitter.
You can make reduced loads in the .458 win that pretty much duplicate .45-70s; I'm not so sure that you can find as much in the way of cheap bullets in .416 as you can for the .45-70.
But -- I don't mean to plug the .458, exactly. I would strongly lean to the .375 H&H -- it is cheap, versatile, and you can find tons of commercial loads for it, and lots of reloading components. The .458 is next in line in my book if you are looking for something heavier. I suppose my criticisms of the .416 shouldn't be taken as criticisms. I've read similar criticisms of the .25-06 (a caliber I'm fond of), which complain that it isn't a .243 win/6mm rem, nor is it a .270 win -- that it is sort of in-between; and I haven't found those arguments to be quite fair (the .257 Roberts is based on the same 7mmx57 case as the 6mm rem, and you can load it to hotter ballistics than the 6mm; you can load the .25-06 hotter than the .257 Roberts, so the .25-06 should be hotter than the 6mm -- if that's what you're looking for, but they are awfully close in performance (can't do a comparison of the .257 roberts -- another of my favs -- at 100 grains with the others, but you can compare 6mm rem, .243 win, and .25-06 at 100 grs at the federal page). So -- I figger there is also lots of virtue in the .416's for similar reasons.
My sense is you're likely to shoot the .375 H&H more, and hunt with it more, than the .416s.
Dan