Why do faculty use two different email addresses? Why faculty usually have two official email addresses, the university and the department address?


Name@university.edu  
Name@department.university.edu


Do they use the second more to communicate with students and the first for other tasks? They could use one and add emails to folders or labels.
As Andy Putman points out in his comment, the two addresses often go to the same place.  Even if there are two different accounts, it's common to forward e-mail from one to the other (or to a third account).  Nobody uses both accounts independently unless they have some idiosyncratic reason to do so.

The existence of two addresses is not nearly as universal as you suggest.  You don't usually see english.university.edu or art-history.university.edu in e-mail addresses, for example.  The departments that do have special addresses (such as CS or math) often ran their own mail servers long before university-wide servers were common.  Some still do, because they don't like the way the university servers are run or don't trust them to be reliable, while others simply provide aliases for the university servers.  Either way, getting rid of the departmental addresses would be disruptive, so they are likely to exist for the foreseeable future.