First of all, here is everything you need to know about the concept of the "Up Button":
Navigation with Back and Up and some of the implementation details: Providing Ancestral and Temporal Navigation.
Generally speaking, the Up button lets you navigate up in the application hierarchy, instead of just navigating back in the application(s) back-stack.
For example, if you work with some kind of app and you get the email notification, you can open the mail client by pressing the notification. Then you can go back to you application you were working with by pressing the Back button ( back-stack ) or you can press the Up button in order to go to the mail client's 'parent' activity ( for example from some EmailMessageActivity to EmailHomeActivity ) to work with the mail client application instead of the initial application ( the back-stack usually is cleared then, so you can only go back with the Back button as far as the the Android Home screen ).
The "Recent Tasks" factor is irrelevant and misleading, it's just another way of starting a new Activity.