I really like the 35 Whelen. I carried it extensively in the Arctic for a number of years shooting everything from ground squirrel to rabbit, ptarmigan, and wolf on the small side to sheep, caribou, and black bear on the larger side. I used a Williams Guide aperture sight on a Remington 700, but only once when I was hurting for food did I shoot as far as 225 yards on a caribou for anything but practice. Now I primarily use an RCBS cast bullet, which weighs in at 261 grains in the 35 Whelen. The cast bullet penetrates better and opens deeper on larger things, and is less destructive on little things. The Hornady bullets I had available then expanded a little quicker than I liked, and for the small stuff I had to make sure I used a head or neck shot.
For 95% of the shooting I did with the 35 Whelen a .243 would have worked about as well. The big 35 does, however, make an impressive stop on a black bear at twenty feet, and on those occasions when I was very close to a grizzly or I heard something grunt in the dark outside of my tent, having something a little bigger in my hands was a comfort.
If you go with cast bullets the 35 Whelen may offer some advantages over the smaller 338-06, but I've never used the smaller cartridge. Realistically, you're talking about the same basic 30-06 case, one with a diameter of .358, the other .338. I doubt that you're going to find dramatic, discernible differences in the field. If I had to choose, I would consider the most likely use. If I expected to keep ugly things off my carcass at close range, I would go with the larger diameter bullet. If I expected to push the range a little on a big game animal under trying conditions, I would go with the 338-06, but either cartridge should do both.