I'd go with the 30-06. Its extremely popular so ammo is easy to find and there are a lot of different types ypu can get. A good box of Hornady SST bullets is about $25-$30. It has a max effective range of 800-1000 meters (super high quality rifles made just for long range shooting, The Remington Model 700's max range is probably closer to about 800 yards for smaller game), and can be used to hunt large game. (not sure about moose and grizzly but anything smaller is probably just fine).
The .338 RUM is hard to find (Cabelas only had two types on their website), It will cost about $76 for a box of 20, It's great for very long range shooting, and is used for very large, very dangerous game.
A 30-06 has a pretty hard kick so if a .338 RUM has a kick that is 4.5 times harder, it would probably knock most people on their butt. Especially if it is their first time shooting a rifle like that and don't know what to expect.
If you plan on hunting very large game or going on african safaris, go with the .338 RUM, otherwise I would go with the 30-06 for a great all around gun.
Hope this helps and don't forget to post a pic.