If we assume the new PSU isnt dud, the board has been damaged. Mobo power comes from 24 pin, so adding that naturally should power the NIC (lights on like you say).
But if you then power the CPU, the NIC stops. That makes me think that one of the controllers on the board isnt working. Adding a PCI 10/100 card may bypass this problem for you, but being honest that board is EOL now and really its about time for replacement.