The op was asking for comments about a woods gun. Having used several calibers for big game hunting including both 30 and 338 I find it hard to believe that more woods hunters haven't found the 338. Those bigger bullets are really efficient game takers.
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This!
Just for chits and giggles I ran the numbers in Strelok as I built a "test rifle" just for doing comparisons like this.
Using Nolser Accubonds:
.308 200 grn at .588 BC muzzle velocity of 2450FPS taken from the Hodgdons site
.338 200 grn at .411 BC with a MV of 2700 taken from a couple different articles
The .308 200 grn does catch the .338 Federal at 450yds, which I guess fits the definition of "catching up pretty quickly downrange" for some guys. By then though both are down in the 1830 FPS where expansion might be iffy depending on the bullet. Woods distances, the OPs intended usage... moot.
Till 450yds the .338fedral has the advantage in velocity and energy. IF we are talking about a rifle for woods hunting, lets say a max range of 200, then at that distance the .338F is carrying 2328ftlbs Vs. 2029 for the .308. Cut back to 100yds (more realistic IMHO) and it's 2748 vs 2363 in favor of the .338F. Go up to a .338 225 Vs a .308 220 and the difference gets greater as the .308 looses more muzzle velocity. Bottom line, due to expansion ratios, the .338Fed will push the same bullet weights faster, whether or not the higher BC bullet even matters is a question of distance.
And IF you're one of those guys that worships on the alter of high SD, then use something like an A- frame or a Partition or a monolithic bullet. Then you can ask yourself "at what point when the bullet exited the animal did the lower SD matter?" because IMHO bullet construction trumps some mathematical formula based on diameter and weight... IF I really sweated SD I'd run around punching holes in animals using solids.
For all around, I'd go with a .308, but if I had those bases covered and I wanted something to thump animals up to the size of elk in a woods setting the .338Federal is a great cartridge. Going with an 8" TGT (reasonable for elk) and a 225 grn it's got a MPBR out to 275 yds and carries 1500 lbs out to around 450yds. Not too shabby for a short action woods cartridge.