Greater range of factory offerings in .308, and if you can't get it anywhere they don't actually sell ammunition. .243 is usually available, but often with less options than the .308.
With the right pill positioned well .243 will drop anything, but .308 with the right pill drops anything betterer with a bit more tolerance for poor placement (although if you shoot the toes neither will kill). I've seen a 150gr .308 break up on the wallow mud shield on the shoulder of a stag, wide shallow wound and a lost animal. But a .243 even with a top grade bullet would not achieve anything better there - going to a stouter 165gr .308 pill by comparison has returned zero failures to date.
16" barrel is fine with .308, but make sure you get a suppressor and learn about caring for suppressed rifles...
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