Remember, there is a whole family of RUMs , Remington Ultra Mags, all based on the same case, just necked down for different bullets, 7mm, 300, 338, 375.
now the 338 RUM and the 338 Lapua are nearly identical ballistically, so then you got to decide if you like what you get when you step down to a narrower bullet.
For one thing, simple piston dynamics, a wider piston base to apply pressure to is more efficient than a smaller base. For bullets, this means a wider bullet has some efficiency gains. However, they are very small. On the flip side, you can get that smaller caliber bullet up to higher speeds, because at the same ballistic co-efficient, it is lighter and hence will fly faster, but that doens't mean it will retain more energy. Now, in theory, you could get a 30 caliber bullet as heavy as a 338, that would have the same energy at the same speed, and with the better ballistic co-efficeint, would loose less velocity as it passes through the air....but of course then you run back into the wider base of the 338 gives you some effiencey gains there.
So you got to decide what is right for you.
here's a nice little chart I found sorry the lines don't line up
.338 Line-Up: Case Capacity
Cartridge Case Length Case Capacity
.338 Win. Mag. 2.500" 86.0 grs.
.330 Dakota 2.540" 98.0 grs.
.340 Weatherby 2.820" 106.4 grs.
.338 Ultra Mag 2.750" 113.4 grs.
.338 A-Square Mag. 2.850" 129.0 grs.
.35/.404 (wildcat) 2.850" 114.9 grs.
.375 H&H Mag. 2.850" 95.0 grs.
.338 Performance Comparison
Cartridge Velocity Barrel Length
.338-’06 (wildcat) 250 gr. 2,500 fps 24"
.338 Win. Mag. 250 gr. 2,650 fps 24"
.330 Dakota 250 gr. 2,900 fps 24"
.338 Rem. Ultra 250 gr. 2,860 fps 24"
.340 Weatherby 250 gr. 2,940 fps 26"
.338 Arnold 250 gr. 2,950 fps 26"
.338 Lapua 250 gr. 2,970 fps 24"
8.59 (.338) Titan 250 gr. 3,100 fps 27"
.338/.378 Weatherby 250 gr. 3,060 fps 26"
.338 A-Square 250 gr. 3,120 fps 25"
On a side note the 300RUM is NOT the same as the 300 win mag. I am not sure what hemicuda meant, but I want to clarify that. Yes, a 300 winmag and 338 winmag are nearly the same (almost the same situation above, of 2 bodies, just different heads, but actually 1 case is slightly different, having to do with a wildcatter beating winchester to the punch)
Also, the 300 H&H in factory loads falls inbetween the 300 winmag and the 30-06, and it is an older round. However, it is a good sized case, so with handloads and a modern rifle you can get right up there within spitting distance of the 300 winmag.