The 340 doesn't recoil to bad. The 338 rum is a beast, the 338-378 will be braked? Another thing, the 338-378 would be neat in a 30 inch barrel heavy bench rest competition, and if it shoots the 300 grain Sierra Matchkings well, you'd have a good rifle.
But, in the 26 inch barrel the 338-378 comes in, save your money, get a 340.
Why? It's between the 338 run and the 338 win. Its not as absurd as the 338-378. You might be able to shoot the 340 well enough off the bench to get good groups (off hand, 3 inch 12 gauge rounds kick worse).

Depends on your experience in shooting heavy recoiling rifles, and what you want to use it for. I strongly considered one of the 338-378 some 20 years ago, but ended up with a 30-378 custom instead. I also ended up disappointed because compared to my 340, i didn't gain much and had smaller magazine capacity to boot. The 340 shoots 210 Noslers at just over 3200 fps. The 30-378 gave very little more energy. I suspect the 338-378 would give another 200 fps over the 340, but shoot barrels out quicker and is a heck of a lot of trouble.

If you can handle the recoil and have a use for it (even if it's just for fun), then go for the bigger one. If not, the 340 is plenty adequate for all North American game. If you shoot it well, pick a good bullet, and stay within a reasonable distance, it will be plenty.