If you want to shoot suppressed, or use a short barrel, or shoot at close-range targets (<100 yards) 300BLK is definitely better. 5.56 is better if launched from a 16" or longer barrel at targets >200 yards or so. Between 100 and 200 yards, it's a toss-up.
Suppressed 300BLK is about as loud as a suppressed 22LR. The loudest noise is the bullet hitting the berm.
300BLK uses faster-burning powders, and it reaches max velocity in less barrel than 5.56. An 8" 300BLK has a lot more muzzle energy and a lot less muzzle flash, blast, and concussion than an 8" 5.56.
Given comparable energy levels, a .308 diameter bullet is going to put more of a hurt on a hostile target than a .224 diameter bullet. The disadvantage is that a larger-diameter bullet has more drag and will lose velocity faster than a smaller-diameter bullet. The larger bullet will also have more arc in the trajectory. So 300BLK is better at short range, and 5.56 is better at long range.