There are no flies on the .350 Rem mag, but I use and love the .35 Whelen because any rifle that is set up to properly feed the 30-06 will work as well with the Whelen whereas a few .350's have feeding issues with the short, belted case. It is slightly easier to reload a beltless case, and heavier bullets are a bit easier to load in the Whelen with normal magazine length. If you are going to shoot 200 gr. deer bullets, there is no practical difference in finding a magazine-appropriate load length.
As for the 338-06, it is probably a slight bit better multi purpose big game round for the one gun hunter, but for me the .35 is best as a specific BIG big game round. I feel the Whelen is at its best with 250 grain bullets.
It's easy to do a search of current manufacturers, I believe both Rem and Ruger have both .350RM and .35W in their lineup. It is also a simple matter to re-barrel any rifle previously chambered for the 30-06 family to either 338-06 or .35 Whelen, and re sizing 30-06 brass is also a cinch. However, the .350 RM be-barrel or brass re-size is not so easy. The advantage of the .350 is the handy short barreled, and short actioned rifles it has been chambered in.