You're gonna be paying about the same amount for the ammo but the ballistics on the 338 are amazing. 1000 yards and you're still doing over 1700 fps and pushing just under that for energy. To put that in to perspective, .45 factory loads peak at around 1100 or 1200 fps and right around 600 ft-lbs of energy AT THE BARREL.
.300WSM is impressive too, just not quite as amazing as the .338.
++on the 308. It will reach 1000 yards. If you want something flatter shooting but with less recoil than a 308, look at the 260 Remington. The great thing is if you already have a 308 all you need is a barrel swap, the bolt and everything transfers over. Also you can use 308 brass when reloading, just neck it down. The .260 is gaining popularity in the tactical shooting world. The only problem is you have to be a reloader for it to be cost effective.
if you ever notice folks that shoot 1000 yds may start doing it without reloading. but its not long before they start to reload for better preformance. there is just to much that goes into shooting long range to get maximum preformance from factory ammo.
Some of you answer depends on what you are using it for. If you just want to make a 1000 yard shot once and a while, pick whatever. If you are wanting to compete, you better check the rule book. I believe that NRA Long Range competition (1000 yard is limited to .30 caliber. I shot both 7mm Remington express and 6.5 X 284 rifles for that. Palma competition (800, 900, 1000) is limited to the 7.62 X 54 round with a 155 grain bullet. It takes a long barrel to get the velocity to keep that little bullet reliably supersonic at 1000 yards. Part of the trick is also that the rifle has a weight limit, so you have to have the barrel length and weight standard when you build a Palma rifle. I never shot the .338 Lapua, but I have shot a .338, and I sure would not want to shoulder that cartridge for a match where you shoot 45 shots for record plus sighters in a day. My sholder hurts just thinking about that.