Although Aircraft Maintenance Engineer or Aeronautical Engineer both are associated with the field of aviation but there is marked difference between the two.
Aircraft Maintenance Engineers are trained and qualified to inspect an aircraft for any kind of snags and solve if there is any. It basically handles the maintenance work of an aircraft, including its engine, electrical and communication system etc.
An Aeronautical Engineer or an Aerospace Engineer basically deals with design, manufacturing and testing of an Aircraft, it engine or other components.