Everything above is correct, however what is missing is the reason for the distinction.
ASME "SA" materials are permitted for use as components for ASME pressure vessels...... for example, one would expect to see SA105 carbon steel components on a vessel drawing.
ASTM "A" materials are permited for use in a Power or chemical process plant piping system designed to ASME B31.1 or B31.3.
Material Records must be maintained by the vessel fabicator for all "SA" materials......... the piping system fabricator must only supply evidence, in the form of a certificate of compliance, that the materials meet the "A" requirements.