Why faculty usually have two official email addresses, the university and the department address?
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.
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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.
While it is possible that some departments are running separate mail servers and faculty members are not linking the accounts, in the cases I am aware of this dates back to times when email and computer logins where managed at the department level and there was no concept of a university email account. Some departments I have been in are actively phasing out the departmental alias such that new staff members do not get a departmental alias while others are continuing to give out departmental aliases. In all cases I am aware of, both addresses link to a single email account.
As to why a faculty member may give out one address over another, that comes down to personal preference.
In addition to the given reasons, I know of two other common reasons why you may often see multiple addresses:
Why faculty usually have 2 official emails, the university email and the department?
Sometimes, the answer is simply "because they can". At various universities I am acquainted with, any employee gets several (more than 2) e-mail aliases for their one university e-mail inbox.
This is in response to the OP's request in a comment to elaborate on my personal situation. This is not to explain why I have two email addresses, but just how I use them differently.
In my department (math), we have separate department and university servers both with email, and I keep my email accounts separate, though many people in our department forward from one to the other (usually from department to university, I think) and only check one.
Basically I use the department email for anything involving research or teaching or quasi-personal, as well as administrative stuff at the department level. Various factions within the university send out loads emails to us (often requiring no response and rarely urgent) which typically go to our university email addresses.
Therefore I made the conscious decision to not forward from one account to the other because almost all of these are unwanted emails, and I probably only open about 5-10% of them. So they would just clutter up my department inbox and distract me with email notifications throughout the day if I forwarded them to my department email, which I check often. I know I could filter the forwarded emails into a separate folder, but I don't want to know when I get a new university email, and having this extra login barrier helps reduce distraction.
So whenever I give my email address to someone, I give them my department email which I check throughout the day, whereas I usually check the university address about once every day or two. I honestly don't even know what my university email address is. Any email I send of my own accord will come from my department address. The only emails I send from my university account are replies to emails to that account.
I would think there is probably multiple departments trying to provide this internal mail service for the university.
The typical case is like there is a department-level IT infrastructure with its own mail addresses and its own policy, while there is also a central IT mail service provided by the university.
The second can be needed, for example if the department has needs which are contradicting the policy of he university (for example, if the Uni wants to give mail addresses only internal employee, while the department want to give mailboxes even for guest researchers, too).
Or the department-level mail address (and password) could be also used for authentication, f.e. to use the department-level computing infrastructure.