I've got a .338/06
The .338Fed is about 90fps slower with equivalent loads. Not enough to debate over. The .338Fed would be preferable if in a "mountain" type rifle, but only if said rifle is under 7.5lbs w/scope vs other options such as an '06.
The recoil is going to be a bit more than the '06, about like a .300mag. in equivalent firearms. The .338 is a level step up from the .30/06 and is on par or better than the .300mags for really big game. I would prefer it to any of my .30's for the large bears. I prefer it to the .338mag due to a 2rd larger magazine capacity 5+1, vs 3+1.
Trajectory is very similar to the .30 caliber weight equivalent. Not enough difference to be a handicap for the larger bore.
I really, really like my .338, but then it dosen't do anything that either the '06 or .45/70 won't do.
I have a 7mm-08 in a Remington Mod7 that gets far more use than the .257Roberts(very light M98 Sporter, .257WbyMag, .30/06, .338/06,.300RUM (too much of a good thing....)and .35Rem. Only the .45/70, .30/30 see equal use to the 7mm-08. Primarily due to handling qualities of the Mod7.
I wouldn't trade those two rifles (yours) for the .338Fed. That is unless you absolutely have to have the Kimber............. Keep the '06 !!!!