The 400 claim is a little high. Specs for both indicate a little less than 300 FPS increase with the 300 grain bullet for the WBY over the H&H and that is significant.
From COTW, 12th edition:
.375 H&H, 2 Factory loads listed
-270 gr., 2,690 FPS/4,340 FPE
-300 gr., 2,530 FPS/4,265 FPE
.378 Wthby, 2 Factory loads listed
-270 gr., 3,180 FPS/6,062 FPE
(+490 FPS/+1,722 FPE)
-300 gr., 2,925 FPS/5,701 FPE
(+395 FPS/+1,436 FPE)
But is it needed? Is it worth the extra cost and trouble of dealing with an exotic cartridge?
That's a different story. I had wanted a .375 H&H from about age 12, but when .375 RUM came out, I decided I'd rather have a rifle capable of spitting out 300 gr. pills at 2,970 FPS (which it does with my loads). Doesn't mean the .375 H&H would have been inadequate, I just liked the idea of a .270 trajectory with twice the bullet.