Vic has the solution! Like that.

They are both good rounds perhaps for different setups, but same size game, just different ranges, it's the BULLET that does the killing, not the cartridge. Only thing I'd say, is make sure what bullet is used in the 338 Federal, the range used is within the velocity envelope for expansion. Energy IMHO is very sufficient to 300 yds, the 338/06 another 100 yds, and the Win mag, perhaps to 450 or so, assuming the bullet is steered correctly and that goes for any round. Any shot most normal cartridges start dropping fast after 300 depending on sight in range, So range estimation, wind dope, and shooting skill/conditions i.e. rest all come to play as much as choice in cartridge. Cup core will do fine at 338 Federal speeds but I would be very interested in seeing what the 185 x does in it, vs 210-225s as far as wound channel and penetration as the wt. retention is so high, a mono bullet is not apples to apples w/CC bullet, and it may compare very favorably to heavier conventional bullets w/better trajectory. That said, IF I were standing down a run in with a bear at close range, I'd be happy to have a heavier partition in chamber. And of all 338 rounds, I would choose the 338/06 because I had one, and can shoot it well, and it holds 1-2 more rounds in the gun....so why not in bear country! Give me a claw style Winchester 70, nice Mauser, MRC, or a Sako i.e. L61R and I would be very happy for hunting anything anywhere in the US. 225 partitions using a 4x or a low end variable no more than 6-8x on top end in solid mounts in good synthetic stock, I'd be set.

If I wanted a nice deer/black bear round that is used occasionally on elk, and esp. in timber, a 338 Federal would suit me fine.

On the other hand, if my hunting was largely elk and bear and I roamed alot of open country, one might enjoy the longer range versatility a well loaded 338/06 or 338 Win Mag can provide.

Just have to ask yourself, what hunting style do you have?