EEPROM, also known as flash. Technically flash is read only.
Not sure what point you're trying to make and its confusing some folks.
EEPROM is Electrically Erasable Programmable Read-Only Memory.
Flash is the name associated with a type EEPROM that can be written faster. "written". The read only part is a legacy statement when there really used to be ROMs that could not be written. They then evolved to allow you to erase them and reburn them. They keep blurring the lines making it faster and easier to write. Essentially it is relatively slow read and WRITE permanent memory similar to a hard drive.
In a phone, when folks usually talk about how much memory it has, they're referring to the flash memory. The iPhone has 8, 16, 32GB of flash memory.
You clearly write to it all the time to write music, video, apps, everything. It acts like the hard drive for the phone.
Folks maybe getting confused with RAM. Random Access Memory. This is very fast access memory used while the processsor is running to run programs and store lots of temporary information. When the power goes out, it instantly forgets everything. Cell phone manufacturers almost never advertise RAM specs. However the amount of RAM is the probably the biggest factor in how fast the phone runs. If it doesn't have enough RAM it has to swap part of the programs its running onto the slow flash memory. If it can keep everything in RAM, its much much faster.
The iPhone 3GS has twice the RAM (256MB) of the iPhone 3G (128MB) and greatly contributes to its speed improvement. New Android phones (Nexus One etc) are shipping with 512MB RAM.