As someone who’s had both, and done load work ups on both, I’ll go with the 300 Norma everyday of the week.
Lots to like about it over the 338L, for me anyways. Flatter shooting and less recoil are my main advantages. For me, it meant more accuracy and easier to spot misses.
The 300 Norma brass is not amazing though, some guys have issues with it, and the Norma likes to form carbon rings in the throat, that you have to stay on top of or it gives you pressure. Borescope really helps here to diagnose that.
I have both barrels for my Desert Tech SRS, and haven’t spun my 338L barrel on once in the last 2 1/2 Years, since Insite did up a 300 Norma for it.
338 is a good cartridge too, and would not totally be a wrong answer though either, lots of components available, factory loaded ammo is pretty good, and Lapua brass is quite nice.