It sounds like the bridge in the enclosure, which passes on the power provided by the Firewire or USB cables to the hard drive itself, has a power supply problem.
This can happen due to component aging - capacitors no longer able to keep the board voltage up to snuff, etc.
The external power supply probably continues to work because it doesn't need to pass through the bridge, as it supplies the needed voltage directly.
If this is the case, the hard drive itself MAY be unaffected (yet), but power supply problems can easily result in catastrophic drive burnout, as well, while a failing bridge will probably result in data loss at some point.
Back up everything on that drive NOW.