The difference in the two documents in their applicability is probably that the service bulletin is published as a generic procedure that might be called-up either as a specific check or used in a scheduled maintenance program, and details how the check is to be carried out; whereas the AD specifically requires a check of all BAe146.
The difference between an AD and a Service Bulletin is:
An AD has been issued by the FAA (or other reguatority agency) and has been published in the Federal Registrar which means it is now a law and must be accomplished. The AD does not provide repair or inspection instructions, that is still found in a Service Bulletin, but since the AD calls out the Service Bulletin it too now has to be accomplished. An AD can be issued for one aircraft of a entire fleet, but normally it will be issued against all already delivered, and must be accomplished at a specific time, either calander time of aircraft time or a combination of both. Aircraft still in production can have the requirements of the AD incorporated prior to
delivery.
A Service Bulletin is issued as a product improvment. The operator can choose to or to not incorporate the Service Bulletin. Again Service Bulletins can be issued for one aircraft of the entire fleet. Normally these product improvements are incorporated during production and the Service Bulletin would be effective on all previously delivered aircraft.