Very simply - Welding and Brazing Procedure and Performance qualification are governed by ASME Boiler and Pressure Vessel Codes and Standards, Section IX Committee, while the requirements for welding and brazing in MIL-STD 248 are governed by the military.
If you want more specifics than this you need to perform a side by side document review.
Check with your Contracting Authority. They may waive the Mil-STD-248 in favor of ASME. Several of the Mil-STD's make the provision for using AWS or ASME.
248 is Navy predominately, and the Navy is familiar with ASME.
