The term "weberface" distinguishes the distinct "Browser + Website" method of accessing a service as opposed to other available means. For example, Gmail provides a number of ways to connect to it, including IMAP, HTTP, and a programming API. All are "interfaces", but "weberface" signifies a specific method for clarity.
It also refers the specific "feeling" for applications that are written for browsers only, or may be (poor) copies of a traditional application that have been ported.
"How are you accessing Facebook right now, through your phone?"
"No, I'm just using the normal weberface."
"The application that came with VMWare Server used to be really great, but in the new release they switched to this crappy weberface and now it never works."