PROM (Programmable Read-Only Memory) is defined at manufacture and cannot be changed or erased, while EPROM (Erasable Programmable Read-Only Memory) allows users to define bit patterns and can be erased using ultraviolet light. Unlike RAM, both PROM and EPROM retain data without power. EEPROM (Electrically Erasable Programmable Read-Only Memory) differs from EPROM as it can be erased electrically without UV light. The discussion also touches on the functionality of UVEPROM and how EPROM cells operate using a MOS transistor to trap electric charges. Understanding these distinctions is crucial for applications in memory storage technology.