I have had several 338 Lapua custom rifles and at least one 338 RUM custom. About the only difference I noticed is that the brass for the Lapua was of better quality than I could get for the RUM (at the time concerned). None of my rifles had muzzle brakes and I could load the 300 Sierra to speeds sufficient to make shooting the rifles an unpleasant experience for most (they'd make snot fly, spatter tears onto the inside surface of my glasses and cause headaches) with either chambering. The fastest I've ever shot a 300 Sierra is just under 3150fps from a LONG barreled 338 Lapua but the precision was not too good once I went past 2900fps or so. The furthest I shot the 338 RUM with the 300's was just a few yards short of 1 mile (28 inch barrel) and the rounds were still plenty supersonic (starting speed about 2825 to 2850 fps).
The only problem/quandry for me is that there is match grade ammo available for the 338 Lapua (Lapua and Black Hills manufacture) but only hunting loads available for the 338 RUM. Every once in a while I'd like to be able to just stop in and pick up some match grade "plinker" rounds for a rifle, this can't be done (that I'm aware of) for the 338 RUM.
If you're only considering loading your own mathc grade ammo with no possible need or desire to get match grade factory ammo I'd say get the RUM.