"Why are sawed-off shotguns illegal?" A more important question to ask is, "Are sawed-off shotguns illegal?" The answer to that question is "No." Despite what many people believe, it is, in fact, legal to own a sawed-off shotgun, or, in legal terms, a short-barrel shotgun. The reason many assume that such weapons are illegal is that they are very tightly controlled and regulated by the federal government, resulting in legally-owned short-barrel shotguns (or rifles, for that matter) being a rarity among the general population.
Before the 1930's there was little, if any, legislation regulating the sale, manufacture, and/or transfer of firearms between individuals. Thus it was possible to own a short-barrel shotgun. However, the rise of organized crime began to frighten the public with stories of gangsters injuring and killing civilians, often outgunning the authorities who pursued them. The federal government responded by passing the National Firearms Act of 1934. Originally intended as a weapons ban, it was passed as a regulatory effort impacting two types of weapons: machine guns and short-barrel shotguns/rifles.
The Act defined a short-barrel shotgun/rifle as having a barrel less than 18 inches in length. In order to own a short-barrel shotgun, or to transfer it to another person, one had to pay a $200 tax (a very high dollar amount in the 1930's) and register the weapon. Violating the law resulted in a $2,000 fine, 5 years imprisonment, or both. The law was initially regulated and enforced by the Alcohol Tax Unit (ATU) of the Treasury Department's Bureau of Internal Revenue, the forerunner of today's Bureau of Alcohol, Tobacco, Firearms, and Explosives, more commonly known as the ATF. The ATF still regulates the ownership and transfer of short-barrel shotguns.
This, of course, refers to federal law. Each state has its own laws controlling the ownership of short-barrel shotguns. However, most, if not all, of the states have exceptions for those who own such weapons in compliance with the National Firearms Act of 1934 (currently codified as 26 USC 5801). There are also other exceptions that many states have. For instance, in many states law enforcement officers are allowed to have short-barrel shotguns as are certain members of the military when such ownership is connected with their official duties. There are also exceptions for those who use such weapons solely as props in dramatic performances, such as TV and movies.
Having now established that short-barrel shotguns are legal to own, under certain circumstances, it might be prudent to discuss the reasons why such a weapon might be desired. There are basically three reasons for owning a short-barrel shotgun: maneuverability, concealment, and the "cool factor."
Shotguns are often a weapon of choice for home defense due to having sufficient power to stop an attacker while being more forgiving of shot placement errors than single-projectile weapons like handguns or rifles.
Depending on shot selection (the pellets that are fired from shotgun shells), one can also choose rounds that will likely stop an attacker while minimizing the risk of shooting through building materials (i.e., walls) and injuring or killing an innocent bystander. However, negotiating rooms and hallways with a shotgun can be cumbersome due to the weapon's length. One option is to substitute the stock for a pistol grip, but this may not be the most practical approach as it can make the shotgun difficult to shoot well. The other option is to shorten the barrel. This allows one to retain the stock for improved performance while giving the user the ability to navigate rooms and hallways with greater ease. This is a reason why many law enforcement agencies, particularly those agencies with teams dedicated to entering hostile situations to arrest suspects or rescue hostages, prefer short-barrel weapons.
Another reason for having a short-barrel weapon is for concealment purposes. This is a much more difficult activity as there are likely few, if any states that would allow for the concealed carry of a shotgun, even with an appropriate permit. Of course, this wouldn't prevent a legitimate owner, in compliance with the law, from concealing a short-barrel weapon on their person while they are on their own private property or other private property where they are given permission to do so. Such weapons can be beneficial in several different types of law enforcement activities. For instance, the U.S. Marshals Service's Witness Security program has made use of such weapons during high-risk protection details.
The last reason is the "cool factor." For some people simply owning such a weapon is the end-purpose. They deem it "cool" to own a weapon that nobody else in their social circle has seen, or resembles something they saw in a TV show or movie. While not as practical, it is certainly no less legitimate assuming the owner has complied with any and all regulations.
As you can see, so called "sawed-off" shotguns are legal to own, given the right circumstances. Within that legal ownership, there are legitimate reasons for going through the process to own a short-barrel shotgun.