From a performance standpoint on deer and antelope, not much. There isn't anything a .270 can do with a 130 grain bullet that the .25-06 can't do with a 120 grain bullet, and vice versa.
Beyond that, it depends what other uses one has in mind. The .25-06 is pretty much acknowledged to be the ultimate long range varminter with 87 to 100 grain bullets, but a .270 with 90 to 110 grain bullets isn't too shabby in the varminator department either. Both, however, are a little heavy in recoil and muzzle blast for sustained shooting.
For game larger than deer, up to elk and maybe even moose, the .270 is hands down the better choice. Some will argue otherwise, of course, and some people hunt elk with a .243, but a 150 grain controlled expansion .270 bullet is going to provide much better penetration than will any comparable .257 bullet.
One other consideration is that a .25-06 generally takes a 24-inch barrel to live up to it's performance expectations, while a .270 will do just fine in a 22-inch tube. Put the .270 in a 24-inch barrel, however, and the .25-06 is going to have to really huff and puff to keep up.
