In my limited 3 airline experience, an AOM is an airline specific tailored manual.
For example, the Airbus AFM contains all the normal, abnormal, emergency procedures and limitations for operating an Airbus aircraft, say an A320.
The airline will take the Airbus manuals, and write a tailored AOM for the procedures the airline will use to operate it's A320s.
At my airline, the AOM only contains normal procedures. It separates out the abnormal (gen overload), emergency (engine fire / cabin fire) and supplemental (cross bleed engine start / deicing) procedures into a separate manual - QRM.
My airline also has an FOM, but instead of containing aircraft specific operating procedures - it just specifies general operating procedures that is applicable across all fleet types at the airline, like weight and balance, or diversion planning, dispatch, security, flight crew responsibilities and qualifications.
So the Airbus AFM is available, but not the primary reference for operating the aircraft - the AOM and FOM are the primary reference.
Another example is when picking up a new aircraft from the manufacturer. At my airline we have an approved weight and balance program that is used to ensure the aircraft is operated within weight and balance limits. But when picking up a new aircraft from the manufacturer, the aircraft has not been accepted into the airline's weight and balance program yet. So we have to revert to private pilot weight and balance, weight x arm = moment, and add all of it up and determine a %CG, and look up in the manufacturer's manuals (and not our airline's procedures manuals) the appropriate trim setting. After ferrying the aircraft from the manufacturer to our airline's maintenance base, the aircraft is imported and inducted into our airline's maintenance program, and weighed to be accepted into our weight and balance program, and then added onto our airline opspecs. After which all of our normal procedures specified in our airline manuals (AOM/FOM/QRM/fueling manual/deicing manual/station operating manuals) can be used.
Another example: In the US, American, United, and Delta all operate the 737 and 320 aircraft. But the way American operates the 320 or 737 is different from the way Delta operates their 320 or 737. All the airlines have access to the Boeing FCOM for the 737 and Airbus AFM for the 320. but American will write their procedures for the 737/320 in an AOM-1/2 FM-1. where as Delta has their 737/320 procedures contained in their OM Vol 1/OM Vol 2/QRH/FOM
Bonus animus sit, ab experientia. Quod salvatum fuerit de malis usu venit judicium.