Understand that a .25-06 is a .30-06 necked down to .25 caliber. You end up shooting a lighter bullet with the same powder charge, hence the 25-06 has a lot more velocity. It also shoots much lighter bullets, say from 85-120 grains.
To me it doesn't kick much more than a 270 (which is a .30-06 necked to .27 cal), but most rifles made in 25-06 are varminters, meaning heavy stock and bull barrel. That means they are a pain to walk around with. (I think my Dad's 25-06 weighs in at like 16 pounds with scope, bipod, sling, etc) However all that weight makes the recoil no worse than a 30-30.
I don't know many people who use the 25-06 for big game larger than antelope, though I'm sure the 25-06 can kill a deer. Given the high velocity though, I'd shy away from the 25-06 on Elk. The bullets tend to break up too quickly, and on something as large as an Elk, you'd likely end up with a wounded animal, rather than a dead one.
25-06 is a rather specialized cartridge, and since you don't re-load I'd suggest the .308, as inexpensive american made ammunition is readily available, and even places like Wal-mart are sure to have it on hand.