With paper piecing you can get precision that you can't necessarily get with regular piecing. Particularly, New York Beauty blocks and the like. All those tiny points are virtually impossible to achieve in any other manner.
Some folks also just like the technique for all of their piecing. Again, precision that you (or those who prefer that method) may not be able to achieve any other way. There is no precise cutting. Very precise sewing and then trim after you have sewn.
Personally I don't mind the technique when necessary but do not feel compelled to use that method for 'regular' piecing.