Originally Posted by Kotts

In addition, a headwind hurts your road speed more than an equivalent tailwind helps it.

The power required to overcome aerodynamic drag increases with the square of the speed increase. So, given the same power output, even though your "airspeed" decreases with the tailwind just as much as your "airspeed" increases with the headwind, your road speed is decreased more by the headwind than it is improved by the tailwind.