I have both but have only hunted the .35. It has performed great but the 338-06 would likely have done as well. I only got the 338-06 this year so that's why I've not yet blooded it.
I think the "better bullet selection" argument for the .338 died a happy death over 20 years ago - plenty of great bullets in each these days.
IMO, the advent of Power Pro 2000-MR powder has been a game changer in favor of the 35 Whelen. Speer has published data for the 250 HotCor at 2700 FPS, and Sierra has published data for the 225 SGK at 2900 FPS. I have not pushed my Whelen this far but 2650 and 2850 FPS respectively are easy to hit without the pressures even breathing hard. This represents a good 100-150 FPS advantage over the .338-06 at the muzzle, and that larger frontal area previously alluded to. The .338-06 retains an edge in BC and SD but many argue in the age of monolithics SD is not as valid as it was in the days of cup and core. Either way they are kind of peas in a pod - darned awesome peas at that. I'm hoping to hear of a new uber-powder that will do for the .338-06 what PP 2000 MR does for the 35 Whelen. I have been playing with PP 2000 MR in the .338-06 and am getting 2700 FPS with the 225, but with what seems to me a touch more pressure than I experience getting 2850 FPS with 225's in the Whelen.
And now I'm about to muddy the waters with my new rebored 9.3x62, in which 2000-MR seems to be a winner too. Medium bores are fun.
Enjoy whatever you choose,
Rex