IMHO the difference for big game hunting is that winning the Weatherby Award when that was a big deal was more likely to happen with the Weatherby cartridge in a Weatherby rifle.
Beyond that no practical difference except price and availability of cases.
That said given a choice between two otherwise identical rifles I'd take the Weatherby and I still wonder at the long ago purchase in .300 Win Mag by somebody else of a Colt Sauer which was available in either and fairly heavy as such things go. Maybe it was a choice of nicer wood.
