338s hit with more authority than 308s or lesser calibers on larger game animals.
They all will kill. I believe the 338s disable large animals quicker, from what I've observed in hunting moose and brown bears.
The lesser calibers will kill them. Generally don't anchor them as emphatically.
For deer-sized animals, I've not observed any difference between the lethality of 7mms, 308s, or 338s.
Have killed large animals with 338WM, 338 Imperial Mag, 338 Edge, 338/378 Weatherby, and 338 LAI.
338WM fine for shots to ~500yds. Killed one bull moose at 760yds with a 250gr Nosler Partition. A round nose at that, back in the mid-80s. Required a LOT of hold over. Took a good while for that chest shot bull to expire. Laid down pretty quickly, but was still alive 20 minutes later.
Past 500yds I'd go 338 RUM on up. Never cared for the 338/378 Weatherby, but a lot of that was due to the soft/weak cases.
I favor the 338 LAI, for longer yardage. But the RUM and Edge are nipping its heels, running neck to neck out to ~800yds.
If you're gonna hunt the larger animals very often, would be nice to have a 338 in the RUM on up Class. Place an effective brake on it, to reduce recoil, so you can shoot it comfortably (well).