The sawn-off shotgun is a modified version of the already deadly shotgun. It weighs less and has a larger "hit angle" than the standard shotgun. However, its drawbacks is that its shells do less damage and that it has slightly less range, but most players agree that the sawn-off is a better choice.
In combat, shooting a sawn-off shotgun has a wider "hit angle" than the standard shotgun. While this may not seem important, it is. The "hit angle" determines how many zombies get hit if you are surrounded by zombies, and when fighting in close quarters (thats what a shotgun is for), the hit angle is very important. The greatly reduced weight (25 pounds to 15 pounds) will allow you to carry more materials and more ammo and is more "scavenging friendly" than the standard shotgun. The decrease in damage is a definite drawback, but even if the shot isn't fatal, it will knock the zombie back to give time until the next fatal shot. The range won't really matter, the shotgun is meant for close range anyway.
Put a saw and a regular shotgun in your inventory, and you get a nice sawn-off.