They are quite different cartridges. The 257 Roberts(a.k.a
Bob, 257 Bob), is based on the 7mm Mauser, which is based on
the 8mm Mauser. The 257 Weatherby is, well, based on the
Weatherby style, with elliptical shoulders, and a big old
long case for a bunch of powder. It probably has a parent
Weatherby case, but I am not much of a Weatherby expert.
The Roberts is a mild mannered efficient cartridge, and the
Weatherby is massively overbore. The Weatherby is at the
top of the velocity heap for .257 caliber. Which is pretty
typical for Weatherby cartridges. The 257 Roberts is a
cartridge that should have been what the 243 Win is today,
because it does everything just as well as the 243 Win,
and it is an improvement over the 243 Win, in deer sized
game. But because it wasn't a 257 Win, or a 257 Rem, it
never got the marketing it should have. I have even read
accounts of old time elk guides using it, but this would
require a skilled shooter, to make it a humane choice for
elk.
Someday I will own a rifle chambered in 257 Roberts. I
doubt I will ever own a 257 Weatherby. I do own a couple
of 25-06 Rem. chambered rifles, and that is as "overbored"
as I need to go in .25 caliber.
Squeeze