I was under the impression that flash memory was another type of EEPROM
Yes, and no. The 32k of "flash program memory" in an ATmega328 is not usable for writing data, and has a limited number of cycles when being written by anything. The EEPROM is much more easily writable, but is only 1k. An SD card offers ridiculous amounts of storage, at the expense of complexity (a filesystem and whatnot.) (If you use the arduino environment, a lot of that complexity may be hidden by pre-written libraries.) In between, you can find external serial EEPROM and "Dataflash" chips that have intermediate amounts of storage, and intermediate complexity...