The ^2 gives the normal distribution, which is special for many reasons. With ^1 you get the Laplace distribution. This is widely used when the data is sparse

The ^2 gives the normal distribution, which is special for many reasons. With ^1 you get the Laplace distribution. This is widely used when the data is sparse (has many elements exactly 0). This is less sensitive to ourliers and in image reconstruction it gives sharper images than ^2.