The advantage to the African, Asian, Australian, hunter is that the large case capacity of the Rigby, Weatherby, and .404 Jeffery (Ultramag) reduces the incidence of excess pressure in hot conditions, without the load becoming anemic. While the new temperature stable powders have gone a long way to addressing that issue, not everyone uses them. The advantage to the North American hunter is velocity as we as a group worship the "Velocity God", and nothing produces velocity like large case capacity. My Rigby load drove 350 gr X and Mag Tip bullets well in excess of 2850 fps, without bursting at the seams, providing .30/06 trajectories, and devastating up close power which is comforting when the rifle is used in the protection role. Comparably, the .375 H&H or Ruger, which are powerful cartridges in their own right, get 2850 with 260-270 gr bullets.
Remington's .416 has proven itself in the game fields, and has been used by successfully against lions by PHs of no less fame than Ross Seyfried. There's even one example that accompanies tourists on coastal hikes around here chambered in a Brno 602. But when compared to the large case .416s, the Remington/Ruger versions are like comparing a .308 to a .300 magnum. The .308 has proven itself in the game fields as well, yet there are those who want a .300 magnum if they carry a .30 caliber rifle.
When it comes to protection rifles, if I were to buy another .416, it would probably be a .416 Ultra (Dakota) to take advantage of the inexpensive brass compared to the cost of Rigby or Weatherby brass. I have nearly 400 pieces of .375 Ultra brass and the cost of acquiring it was far less painful than it was to acquire 200 grounds of Norma .416 Rigby brass at $180/50 in the late '90s. Another advantage of Ultramag brass is that more rounds can be fitted in a magazine than is possible with a Rigby or Weatherby chambered rifle, without altering the magbox.
My .375 Ultra compares favorably with the .416 Remington or Ruger; we see that I can drive a 350 gr bullet (SD .356) to 2350 fps from a 20" barrel where the .416s can drive a 400 gr bullet (SD .330) to 2300 from a 20" barrel. But a big case .416 will drive that same 400 gr bullet 2500 fps in the case of the Dakota and 2600 in the case of the Rigby/Weatherby when shot from short barrels. An increase of 300 fps represents an advantage for the hunter who can make use of it.