The .338 is a fine cartridge for someone who hunts moose and big bears. Many find the recoil objectionable, but often this is from people who are not used to heavy recoil, or whose rifles are too light. The .338 shines on heavy game, but it is not as versatile as the .30/06. Under most conditions the .30/06 will handle any North American hunting scenario, but if following a wounded grizzly into the willows is in your plans, the .338 isn't enough either. They don't make guns big enough for that game.
I think most folks would find that in circumstances where a .30/06 or .300 magnum wasn't enough, they would discover that rather than a .338, a .375 of some flavor (H&H, Dakota, Ruger, Steyr, Ultra, or Weatherby) is the next logical step. If someone wants to chime in about the 9.3X62, then yes that works too. I believe these cartridges produce real world terminal ballistics in terms of depth of penetration and wound volumes that are are a measurably greater than those produced by the .30/06. We sort of run the risk of an apples and oranges comparison here, but if each cartridge is loaded with a bullet weight that allows the cartridge to produce 2500 fps at maximum pressure, there is a greater difference between the .375 and the .338 than there is between a .30/06 and the .338.
