A background task is just something that's computed outside of the standard http request/response process. So it's asynchronous in the sense that the result will be computed sometime in the future, but you don't care when.

Distributed just means that you can have your task processing spread out across multiple machines.

A specific example would be, let's say, after your user registers on your website for the first time you wan't to get a list of all his facebook/twitter friends. This action will take a long time and is not vital to the whole registration/login process so you set a task to do that later, and let the user proceed to the site and not make him look at the spinner the whole time, and when the friend list becomes available it will show up on the website (on his profile or whatever). Makes sense?