They all do the same job, and are within about 100 fps of each other; but two of them are more for "bragging rights" or reproductive organ compensation:
.300 RUM - expensive ammo; expensive brass; unnecessary recoil, muzzle blast, and erosion. A very inefficient cartridge.
.300 Weatherby - expensive ammo; moderately expensive brass; ridiculous recoil, muzzle blast, and erosion. An incredibly inefficient cartridge.
.300 Win Mag - affordable ammo; affordable brass; tolerable recoil and muzzle blast; less erosion. Inefficient, but justifiable.
The .300 Win Mag is the reasonable limit of .30 caliber cartridges.
It's like the Bugatti Veyron....
The original version only used 266 hp to reach 155 mph, but requires another 721 HP to reach 253 mph. (for 987 hp)
The SuperSport model has 1,200 hp, but that only increases top speed to 267 mph.
The newest model, the SuperVeyron, requires an additional 400 hp for a measly 21 mph increase in top speed (for 288 mph and 1,600 hp).
The .300 Weatherby and .300 RUM are the "SuperSport" and "SuperVeyron". They have to increase chamber pressure and powder charge significantly, to barely surpass the .300 Win Mag's ballistics. To me, the impact on my wallet and barrel life (another hit to the wallet), just aren't worth it.