From my experience, yes, a 338 offers more than the 300.
After seeing and witnessing elk killed with 243 to 375, being in Africa last year with 5 other guys with a 300 H&H, 330 Dakota, two 338's, two 375's and 30 some animals taken from steenbok to eland.
My subjective opinion is that bigger diameter bullets hit harder. All the other calibers worked just fine, but for visible impact, the bigger calibers have it. That line starts at 338 and bigger in my opinion. YMMV
