First, flash memory is a type of EEPROM (electrically erasable programmable read-only memory), so comparing them either/or doesn't make sense.
Second, there is no hard definition of "flash". It's more of a marketing term. It has converged to generally mean a type of EEPROM that is optimized for density, typically at the expense of slow write and erase times, and large erase blocks.
So given the above, flash memory is generally used for storage of large amounts of data. Those USB memory sticks hafe flash inside. In fact, they are sometimes called "flash drives".
In contrast, smaller EEPROMs that can erase and re-write individual bytes are often used for relatively (compared to a file system) small amounts of storage for things like configuration parameters, calibration coefficients, and the like.