Should a student use the email account provided by the university to communicate with professors?

In the cases I'm familiar with (U.S. universities), using your own e-mail account should be completely fine, subject to some obvious caveats. One is that it's best to have an e-mail address that doesn't look foolish or offensive. People sometimes choose very strange usernames, and you don't want that to reflect poorly on you; furthermore, you should make sure your e-mails include the name your professor knows you under. Another issue is that if you are asking for sensitive information such as grades, your professor will likely be unwilling to send this information to an outside account without some verification that it belongs to you. Finally, you should make things simple for anyone you correspond with. For example, if you use several accounts, you should check them frequently or forward one to the other, so that there are no delays if someone sends something to an account you didn't expect.

One common solution is to set up your university account to forward to your private account, and to set up that account so you can send e-mail listing your university account as the sender/return address when necessary.

In the UK, universities are very strict about information they will release -- for example we would not tell someone which courses a student was on.

This means that any email that comes from a non-university account must be treated carefully -- if by replying I appear to acknowledge the sender's name, and that they are taking my course, I have just illegally leaked confidential information.