Maximum allowable working pressure (MAWP) is the maximum pressure that the weakest component of the system can handle. This is based on design codes and back calculation from the fabrication of the vessel, piping, etc. The MAWP should be clearly stated for vessels or piping to figure out what pressures the system can withstand. Note that the MAWP does not remain constant throughout the life of the system and will reduce due to corrosion (carbon steel), wear and fatigue.
Design pressure is the maximum pressure that the system that can be exposed to and sets the system relief valve at the same pressure. This should be below MAWP and based on company standards can vary from 10% to 25% above the maximum operating pressure of the system. Maximum operating pressure is usually the high trip pressure of the system.