The parent case of the .338 Lapua is a .416 Rigby. Basically a .416 Rigby necked down to .338 diameter. The .338 Lapua was introduced in 1989 and is a rimless bottleneck type cartridge. It measures out to 8.58X71mm. The base diameter is 0.587".
The parent case of the .338 Win. Mag is the .375 H & H Magnum. Basically a shortened and necked down .375 H&H Magnum cartridge. It was introduced in 1958. It is a belted bottleneck type cartridge. It measures out to 8.58X64mm. The base diameter of this cartridge is 0.513".
The .338 Lapua is primarily a military long range round. However it has been used for hunting larger dangerous game (usually Afirca) where this smaller caliber is allowed. Certain African countries have a 9.53mm or 0.375" minimum bullet diameter rule.
The .338 Winchester magnum is primarily a hunting round used mostly in North America and is fairly common for the larger North American game.
This ammunition will not interchange with each other as you can see by their dimensional differences. Also, they perform quite differently from each other.
As for the rifle, the original BAR had detachable magazines and had a full auto feature. Pretty much a military only weapon. The modern day BAR's do not have detachable magazines. They are fixed and hold only a few rounds as to be legal capacity for hunting. They also have been changed up internally so as they cannot be made so easily to full auto mode. They come in quite a few different calibers in long action and short action from .243 Win. to .338 Winchester Magnum.