An extended Application class can declare global variables. Are there other reasons?
Offhand, I can't think of a real scenario in which extending Application is either preferable to another approach or necessary to accomplish something. If you have an expensive, frequently used object you can initialize it in an IntentService when you detect that the object isn't currently present. Application itself runs on the UI thread, while IntentService runs on its own thread.
I prefer to pass data from Activity to Activity with explicit Intents, or use SharedPreferences. There are also ways to pass data from a Fragment to its parent Activity using interfaces.