An ApplicationExtension will be initialized when the mobile application initializes, and will be uninitialized when the mobile application uninitializes. In other words, it will be loaded and effective throughout the whole life cycle of the application and not
be affected when mobile projects are loaded/unloaded. This is the key difference between ProjectExtension/Task and ApplicationExtension.

If you need to manage resources (i.e., hardware devices) that are not dependent on mobile projects, you can create an ApplicationExtension. A good exmaple is a customized Gps management extension that implements different UI and functions for certain Gps venders, and need to be used by all projects. Or if you have a bluetooth camera or USB rangefinder that you want to use with the mobile application, you can manage its settings with an ApplicatinExtension. Another exmaple could be a login window where user needs to provide credentials before they can use the applicaiton. If you have other use cases that manage settings at application level, ApplicationExtension should be an option.

System.Object
   ESRI.ArcGIS.Mobile.Client.Extension
      ESRI.ArcGIS.Mobile.Client.ApplicationExtension

Reference