I've handloaded for both and I've seen the .243 kill lots of stuff. I've also seen the big cased 6 mm's and the big cased .25's, like the .240 Wby, the .25-06 and the .25-284 kill lots of stuff at the longer ranges.
Like Mule Deer says, if there is a difference, I haven't seen it. With premium bullets, they will amaze you as to how well they can kill.
The big difference to me is that the Roberts gives one 2900 fps. with any 100 gr. bullet at much lower pressures than does the .243. The report isn't near as tough on the ears, and the brass seems to last however. Much easier to find accurate loads as well.
While ammo for the .243 is easier to find, even premium .257 Roberts factory ammo has always been availiable, even here in Kalifornia.
I'd much rather have a .257 Bob if I had a choice. E