The Display Name is the name you have given your created email or document template and is the name that is visible in drop down menus. The Display Name is only used for you and your staff and is shown as the Admin Display Name.  

The Client Visible Name is what is displayed to the client as the subject upon receiving the email or as the name of the document they are receiving in the email.  

Example:  Say you create an email template that is scheduled to be sent 3 months prior to a wedding date.  You could give it a Display Name of "3 Month Wedding Email".  However, in the Subject line of the created email you can call it "Your Wedding!"  This is the Client Visible Name that would appear as the subject as your client receives the email.