These profiles and shapes are generally interchangeable between cornices, headers (hoods) or crowns. Some people even switch the names using crown for cornice or header for crown. The shapes are usually bigger than casing or banding and usually have a larger base and corresponding larger top.
For Suncoast Stone we use the following definitions, and as you can see they cross over.
Cornice – An ornamental molding running round the walls of a room just below the ceiling.
Crown molding – The molding forming the top member of a door or window usually when there is minimal or no casing already existing.
Header – similar to crown molding however usually goes on top of a substantial existing window or door casing.