The 338 Winchester Magnum. From what we've seen hunting, the Winchester rounds are more inherently accurate, easier to shoot, as well as less prone to barrel wear and other problems. You'll find the Winchester ammunition to be cheaper and more commonly available. The rifles chambered for the Winchester will be less expensive as well. When you hang "Remington Ultra Mag" on anything, the price doubles. You can buy the Winchester ammunition for $30 a box, and find it in most places that carry a lot of ammunition. The Remington is most often $60 a box, if you can find it.
A 300 Winchester Magnum is more than enough for elk at 200 yards or more, especially with a 180 grain bullet, if you can shoot, it'll take elk at 400 yards. About 999 out of 1000 hunters and shooters should never shoot more than 150 yards, at best. And most of them packing the latest high dollar barrel burning hot rod can't hit the ground with their hat.
I won't even consider a shot on game at much over 200 yards unless near perfect conditions exist. You owe it to the animal to do it right. If you aren't going to burn a minimum of 200-300 rounds of ammunition to get you and your rifle ready, including at least 100 rounds at the longest range you'll shoot, you have no business taking those shots. Maybe 1 out of 1000 people will spend the time and money to get good enough, the rest will complain about the price, the lack of places to shoot, and the bruised shoulder.