I forgot to add, my father is looking for a 300RUM, since he likes mine so much, so I have been looking at lots of loads, and ballistics for it. Since you are thinking about either one, I would suggest running the numbers through the calculators to compare the performance. I am not sure what bullets you are going to shoot, so my numbers may be different.
When I ran my loads through a recoil calculator, I came out with 1 ft/lb difference between the 300 and 338. I can tell you, it doesn't feel much different on the shoulder.
Recoil Calculator at HuntAmerica.com
I run some loads through the calculator from Hornady, there are others, this is just one of many, but it will give you info to compare loads for the two.
I came to the conclusion that there isn't but a couple of inches difference between the two calibers, with a slight edge to the 300, but the 338 is right on its heels, and with more energy at longer distances.
Ballistics Calculator - Hornady Manufacturing, Inc