I have both. I like the .338 more than the .300's I have. Mostly because I find the .300's have a faster sharper kick with 180's. I am trying some 200's this year to see how I like that recoil.
Truth be told I only use the .338 occasionally these days. Almost all my hunting ius with an '06.
I like the .338 for a couple of other reasons too.
First, I believe that I can see more reaction when I hit an elk with the .338 than the .30 cals. It may be my imagination but its mine.
Second, the ballistics of the .338 compared to my '06 is very close. The .300 mags, especially the .300 Weatherby are somewhat different.
