I've shot both calibers and don't think you'd notice much difference in either caliber's recoil if you're just shooting a couple rounds at big game. Both kick and will get the job done.
Having said that, I prefer the 338WM with the heavier 250 grain bullets for elk and moose. They have never let me down. Now if I were hunting brown bears or grizzlies, I would definitely want a 338 instead of a 300.
In the end, it pretty much comes down to your preference. If you can, shoot both. Better yet, if you can, buy both. Then you can study this problem in depth.