You decide what is the difference. Below are descriptions of each.
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.
Minimum pressure is expected during normal operation and may occur in some process transient period or different operating mode and is built into the design for any uncertainties due to start-up, fouled, decayed, etc. It provides some level of flexibility for a proper operation of the system throughout the entire life of the system.