The Hunting Rifle has moderately better damage even after Cowboy, has near-exact pinpoint precision, reloads quicker, has more crit damage, has a higher random crit multiplier, has more durability, and has armor piercing ammo available. Unlike other .308 weapons, it also can withstand extended firing of .308 JSP Hand-Loader perk rounds.
The Carabine weighs one less in the pack, has a lower skill requisite, a lower strength requisite, faster firing rate, and probably cheaper ammo.
If you're on PC, there is a fix for the off-kilter 1stperson non-pistolgrip rifle iron sights. I highly recommend it.