As far as velocity difference between 338-06 and an "improved" 338-06, there isn't going to be much difference. The case capacity of the 338-06 is about 73.4 grains of water. The biggest "improved" case I can find holds 76.7 grains of water, a difference of about 3.2%. Using the 4-to-1 rule of thumb, the increased case capacity should deliver around 0.8% or less than 25FPS of additional velocity using the same pressure and barrel length. Of course, you can push pressures WAY up and get lots more velocity, but...
I currently have a 338-06 Ackley with a 25" Douglas #5 barrel. In hindsight, the hassle with fireforming cases before I can begin to work up loads and the feeding problems from the steep-shouldered and straight-sided case was WAY too much relative to the 25fps benefit and whatever improved case life I'm getting.
If you look at the Barnes manual #3 and compare the 338-06 to the 338 Hawk/Scovill, you will see that the un-improved round is always within 50FPS and sometimes is faster than the improved round. I'd guess that the "improvement" is less than their margin of error for measuring pressure and velocity.
I'm rebarreling my 338-06AI to a shorter, lighter 22" barrel in 338-06 and looking for a little less than 2700FPS with the Barnes 210grain TSX bullet. It will be sesveral poinds lighter than my big boomer and hit harder than my .308 mountain rifle at typical hunting ranges. So, in my case, it fills a gap in the battery.
YMMV