Automated attendants are menu-based dialogs that prompt the caller to press a key and have the system respond accordingly. A main greeting is one example: "Thank you for calling. If you know your party’s extension, please enter it now. For Sales, press 1…"
Call routers are not dialogs, but programs that route a call based on various criteria. A classic example of a call router is time of day and day of week routing. In this case, a call comes into the call router; the call router checks the day and time and routes the call accordingly.