I've shot several brown bears, and was in on many other big bear hunts.
Back when i was looking for my "perfect" brown bear rifle, one choise i was considering was the 350 Rem. mag... (same balistics as the 35 Whelen) I had (3) different rifles chambered in .350 RM includeing a Ruger 77. At that time i also had P.O. Ackley make me a .338-06 (rebored 30-06) in another Ruger 77, and a couple .338 Win. mag's.. After much handloading and useing both extensively on big game with different loads, i came to the conclusion that the .338 bullets "out performed" the .35's on big game, so i no longer used the .350 Rem mag. after that.
Instead i built me a rifle useing a LH 700 Rem. action, Douglas premium bbl. and english walnut stock that's in the below picture. The first (3) years after i settled on the .338-06 i harvested (13) big game animials with it includeing moose, caribou, brown bear, sitka blk tailed deer ect... And although i no longer live in Alaska i still have the rifle.
After haveing said all of that, i do agree that "today" useing NP's in both, they both would be plenty for any brown bear at reasonable range. I said useing NP's because i believe they are the best designed "expanding" bullet for big game ever designed.
For big bears you need an expanding bullet with LOT's of penetration and NP's are designed to do this better than all else. I found that "in general", .338's penetrate deeper than the 35's but i expect the NP's would even this out a bit...
Drilling Man