I believe AS is required for self-joins. A common expository query is to find all employees who earn more than their managers, which can be expressed via "select name from staff as employee join staff as manager on (employee.manager_id = manager.id) where employee.salary > manager.salary".
Also, SQL allows a sub-select to be listed in the FROM clause. But in such cases, the sub-select must be given a name (tables have a name, but sub-selects don't).
As an aside, the keyword "AS" is allowed by not required.