You cannot directly copy a profile file from a Memory Unit (MU) to a Hard Drive (HDD) as a file transfer. The Xbox 360 handles profiles differently than save files.
Do not move the profile file. Instead:
The 20GB drive (original "Phat" 360) works identically to larger drives for profile storage. Ensure it's an official Microsoft drive (or a properly flashed Western Digital Scorpio BEVS) for the transfer option to appear reliably.
Summary: Use the System Storage menu to "Move" the profile from MU to HDD. For new consoles, "Download Profile" via Xbox Live.