I have both. I killed my first deer with my dad's Rem 700 25-06 and I have that rifle now. I also have a .257 Wby Vanguard. Both are sub-MOA rifles. I've killed loads of mulies & pronghorn with both.
I like the .257 Wby, but I love the .25-06. It has less recoil, feeds like a dream and wasn't picky when developing a load. The .257 has tremendous muzzle blast too. It's loud! Both will work. I would use monolithic bullets, especially in the .257 Wby. Velocities are high and you want something to hold together. Im using HVLDs in the .25-06. As others have said, bullet selection for long range is still partly lacking and most factory barrels don't have enough spin.