I believe the use of these adjectives depends on wether you are talking about the process/method/implementation with which something is done, or talking about the actual execution or result of that process.
Automated refers to the process:
The factory is automated
Automatic refers to the execution of the process:
The car production is automatic
While it is harder to apply that distinction for more abstract concepts, it is in general possible to distinguish between a process and its execution.
The difference between "automatically" and "in an automated manner" should then be clear. The "manner" or the "tasks" correspond to the process, and thus are "automated". The "done tasks" are the result of the process, and their execution is "automatic", they are "done automatically".