Two wildly different cartridges for very different uses. The 338 WM is loud, kicks like hell, and is quite suitable for Elk, Moose, and Big Bears. The only thing it could be considered weak for in North America would be a charge by a very large Bear at close range.
The 338 Federal on the other hand is an excellent all-around rifle. Good for everything between Coyotes, Antelope, Mule Deer, and even Moose. A bit light for Elk. On Bears it will do fine up to around 600 pound animals. Anything larger I would want more power.
The 338 Federal is the weak sister of the 358 Winchester. The 358 Win has the advantage in bullet weight, bore diameter, and bullets engineered to the velocity range of the cartridge. If you handload the 338 Federal you must be very careful about bullet choices as most of them are designed to be driven much faster than the 338 Federal is capable of doing.
The 358 Winchester also has the advantage on the small game end with 110 to 140 grain revolver bullets driven at 1800 to 2000 fps. On the Big end the 358 Winchester can handle Bears up to around 750 pounds. The ticket would be the 310 grain Woodleigh at 2,050 fps. This load should be kept down to about 175 yards and under.
The 358 Winchester is currently available in the Browning Lever Action or BLR. This is essentially a rotating bolt action driven by a rack and pinion gear attached to the lever. Clearly the best lever gun in the world and only weighs 6.5 pounds. With a Nikon Monarch Gold 1.5 - 6 x 42mm scope in Warne rings on Weaver bases it goes out the door loaded at 7 3/4 pounds. It is also available in right or left handed bolt actions by New Ultra Light Arms.