As
@Wishfulthinker580 said above, one thing you need to consider is sectional density - the ratio of an object's mass to its cross sectional area with respect to a given axis. This is a representation of how well a bullet will penetrate. Generally looking to have a SD above .300 for hunting DG and the larger the number the better.
Given the same bullet construction (I used Hornady DGX Bonded) the sectional density of the 400 grain .416 bullet is .330 and the 500 grain .458 bullet is .341, so a slight edge to the 458WM. Pushing the .416 up to 450 grains or the .458 over 500 grains will increase the SD of each.
458WM ammo cost more especially if you don't reload...but there are several more bullet options than the .416 if you do.
458WM has more recoil, but not a lot. Are you ready for that? How many times are you looking to hunt DG?
458WM can be reamed out to 458LOTT to give you more versatility. IMO, the 458LOTT is the starting point for a "stopping caliber rifle" if you are truly looking for that. The 458LOTT also make numbers that the 458WM can only dream of.
Lot's of things to consider, but I wouldn't get hung up on the numbers. The 458WM will hit harder than a 416RM at DG ranges because of the greater mass of the bullet and better SD, there is no doubt about that.
Both are fun to shoot and if you just want something big and fun, there's no reason to let this opportunity pass you by. Good luck with your decision.