Swap calibers and this may be the most asked question on this board.
Either caliber will cleanly kill any game animal in North America, within the limitations of the rifle/cartridge/shooter's skill.
It is hard to knock the 30-06, as ammo can be purchased just about anywhere in the world, in any outlet that stocks a bit of ammo. The bullet weight and bullet performance selection of factory loads is very broad with 30-06, an unlimited if one factors in hand loading. While a shooter must expect some recoil, most anyone can adapt to handling the recoil that the 30-06 dishes out. One of the only downsides that I can think of, is effective range, especially on heavier deer like elk and moose. While some would say that they can stretch the range of a 30-06, my opinion is that when you "learn" your rifle and cartridge load, you may limit the shots that you take, on deer/elk sized game, to under 350 yards. This, due to the top muzzle velocity, for which that cartridge can be loaded.
It is in the area of higher muzzle velocity that one can argue for the 7mmUltraMag. The muzzle velocity of the 7mmUltraMag can be pushed to 500fps and more, then the 30-06 can achieve. This higher muzzle velocity translates into a flatter trajectory, which in turn, means that much of the guess work is taken out of "holdover", when making a longer shot. The other drawback attached to the 7mm is an inability to go to heavier bullets. The 30-06 can be loaded with heavier bullets then can the 7mmUltraMag. This may be important, depending on game and terrain hunted.
If forced to choose, I'd opt for the 30-06, due to the broader versatility of that cartridge, over the 7mmUltraMag. However, if you enjoy reloading and really like firing a barrel burner that really shoots hot and using a ton of propellent, get the UltraMag.