Depending on the brass price, affordability might be the main difference. The 338 Norma will fit into a standard long action like the Rem 700 etc. The 338 LM is usually chambered into .50 system due to the length, with corresponding price increase. Ballistics wise, you can probably squeeze a tiny bit more from the LM, but quite frankly, once you start pouring 100 gr+ of powder into a case 5% more doesn't make all that much difference.
I used to love being able to hit hard at 1000 yards. As I get older I find hitting a mini ram at 200 yards with the 22 oddly more satisfying.