EEPROM being both "programmable" and "read-only"
Although they both involve changing the contents of memory, writing and programming are not the same thing.

Writing is done with the chip connected to the processor, during a normal processor cycle, and using normal voltages.

Programming involves conditions that are not normally produced by a processor.  There may be a need to erase the memory first (possibly using ultraviolet light).  It may require higher voltages.  It may be more convenient to remove the chip from the circuit and program it with dedicated programming equipment.  In most cases, the programming process takes much more time than a regular memory access.  The bottom line is that a normal processor write cycle is not adequate to program a chip.


RAM can be directly written by a processor.  There is no need to program RAM.
Masked ROM can neither be written nor programmed.
However, the various types of PROMs can only be programmed.  A normal processor write does not generate the conditions needed for programming, so PROMs cannot be written.