Well, the lamp the other side will get brighter under braking as that's the brake lamp coming on!
Firstly, try a different bulb.
It sounds like the brake lamp in that cluster is earthing through the side light circuit and vice versa and as you have inspected the socket and there's nothing wrong with it then it must be a fault inside the bulb. Don't assume that it'll work properly because it's new.
The only other thing that could cause this has been hinted at above, you do definitely have the right bulb in the right way round don't you? The two pins on the sides should be at different heights if you have the right bulb. This is to make sure you get it the right way round as the two filaments for tail and brake are different power ratings. If you've got the wrong one or forced it in the wrong way round there could be all kinds of shorts.
Actually, I think I may have just worked this out. Have you put in a single filament bulb? It should have two pads on the base rather than just one large one. One pad for each filament. A single filament bulb could well cause this.
Edited by Dan, 24 December 2006 - 12:14 AM.