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Greater mass affords greater momentum, which affords superior penetration, less deviation from intermediate barriers (like glass) and less wind interference. The .338 Lapua is both heavier and faster than the .308. I think the heaviest common .308 load is 180 grains while the lightest common .338 load I know is 200 grain. And nevermind .338 goes about 600 fps faster than .308 (this is an estimated average, some loads of each will go faster, others slower).
Greater velocity affords superior terminal ballistics and flatter trajectory.
Given the .338 has much more of both mass and velocity it will produce much more grievous wounds but also penetrate much further than a .308. Many .308 load have issues with overpenetrating a man sized target, so it’s no stretch of the imagination that a .338 could easily hit a target, overpenetrate, pierce a wall and kill an innocent on the other side. Hell! Depending on the store’s construction, I might even penetrate the whole store and hit someone outside, beyond the store.
(Sorry for wall of text)