Check the voltage at the taillight and then reproduct that voltage on the bence and see what current the module takes, with all combinations -- running, beake, and running+brake (depending on manufacturer/model, the last case may or may not be used).
Then do the same thing with the incandescent bulb you replaced -- the incandescent running light will probably take about 0.5A and the brake light will take about 2 or 2.25A, and together I guess they might take close to 2.75A. It's hard to imagine that the replacement LED bulb could take that current and be happy.
What's the vehicle?