Diagnosing bad grounds is one of the fundamental skills I teach new students when we begin electrical system training. It's absolutely essential knowledge for anyone who wants to be a successful electrical system technician.
The reason the performance is different with the headlights on is the fact that the brake light signal is trying to use the tail light wire as a ground when there's no voltage on it (lights off). Turn the lights on ane the brake signal can no longer find a ground- - - -the brake light goes out alotgether. Check the socket itself for corrosion, and also check the bulbs for a broken filament-to base connection. If the globe can turn in relation to the base a little, the normal ground has broken, and the signal has to back up through the other filament to find a ground. Normally, the filaments ground themselves to the brass base through the little solder joint on the perimeter of the base. Also check the connection of the bulb base to the socket, and the socket's ground connection to the body or bed. The voltage MUST get back to the battery with little or no resistance, so every ground connection from the bulb to the battery must be good. Have fun!
Jerry
