Not to be picky cobalt402, but technically moose are members of the deer family. And a moose can weight 4 or 5 times as much as one of our average, Idaho mule deer.
Nevertheless, there's not a deer in Idaho, including a moose, that can't be successfully taken with a 7mm-08, if you do your part. I'm partial to .30 calibers myself, but my wife, our oldest daughter, and now one of our grandsons have been killing mule deer with 7mm-08s since the cartridge was first commercialized.
That said, I think a .257 Roberts with the right loads would serve a deer hunter just as well as the 7mm-08s have served the other deer hunters in my family. Except for that one thing – as rugerjunkie mentioned, I too think a .257 Roberts requires a long action. So practicality wise, a deer hunter might as well step up to the .25-06.
On the other hand, to me personally the .257 Roberts has a certain "panache." Panache is why when I had my custom, retirement rifle built, I had it chambered for the .308 Norma Magnum. I know my .308 Norma Mag won't do anything a .300 Win Mag won't do. In fact, on paper a .300 Win Mag will do just a little bit more as it's slightly faster for all bullet weights. But everybody and his uncle has a either a .300 Win Mag, or a 7mm Rem Mag. My .308 Norma has panache. And that's why if I personally was going to choose a .25 caliber rifle for deer hunting, it would be a .257 Bob rather than a .25-06 Remington.