Power in a smaller package is the answer. The 10mm had too much recoil for the masses, and it required a large frame gun. The FBI started to download the 10mm, and some bright guy figured out that you could get the same ballistic performance out of a shorter case. The shorter case had the advantage of being able to fit in a medium frame (i.e. 9mm size) handgun. The combination resulted in a weapon more powerful than a 9mm, but with more ammo capacity than a .45, housed in a pistol the size of a 9mm. The California Highway Patrol adopted it almost immeadiately, and the FBI and many other 9mm users went to it.
The .357Sig is basically the same cartridge firing a smaller caliber bullet at a higher velocity. It gives better penetration, and the caliber is still the same as the 9mm/.357mag of police favor. Considering that most police shootouts these days either have a suspect in or behind a car, penetration is a big plus. The old .357mag has always been the #1 one-shot-stop cartridge. This was an attempt to get that performance in a modern autoloader. Like it's .40cal cousin, it fits in the same medium frame guns with the same capacity mags. The Virginia State Police (highway patrol) were the frist to adopt it, and many others have also followed suit.
Both are great rounds, and the obvious advantage to anyone owning one, is they can normally switch to the other if they feel like it. There were plenty of other rounds that have the same, or better performance. It's just that these two combine several other features, like the size of the gun to handle them and magazine capacity, that the others just don't.
Ross