As far as ASME Section VIII Division 1 goes, see definition of MAWP in Appendix 3. This Code does not define or use the term MAP but the concept is the same as the basis for "calculated test pressure" as defined in Appendix 3. Other Codes probably have same definitions.
MAWP ("maximum allowable working pressure") is maximum pressure based on allowable stress at design temperature, reduced for the pressure due to the operating liquid acting on the component or vessel, and in the corroded condition.
MAP ("maximum allowable pressure") is maximum pressure based on allowable stress at ambient ("cold") temperature, it is assumed there is no operating liquid acting, and component is in new condition. This is sometimes referred to as "new and cold".