To initiate email to a customer, click Send Email in the main toolbar.
This is useful because any replies from that customer will automatically be appended to the end of the same case and everyone else on your team can see the entire history of the email transaction.
If the email is related to an already existing case, click the Email link in the case itself. This associates an existing case with an outside correspondent.
For example, if a customer contacted you by phone, and you opened a bug based on their problem, but now you want to correspond with them via email, use the Email button inside the case to start an email correspondence which will be associated with that case.