A smartphone's processing speed affects, first and foremost, the speed at which applications can be opened - and the number of applications that can be run simultaneously, without lag. RAM is also an important factor in these processes. Faster processors also enable smartphones to run certain kinds of apps, so some slower devices will be unable to run certain games, for example. It is important to remember that many smartphone apps these days run off the Internet, and no matter how fast your phone is, the functionality of these apps will be determined by the speed of your Internet connection, not the phone's processor.
