Fan runs when in "on" mode, but not when in "auto" mode while in "cool" setting
Usually A/C, heat, and fan run the blower at different speeds. It's possible that the speed the A/C uses is dead.  

Any easy test, is to swap the motor speed wires connected to HEAT and COOL on the board. Then see if the blower starts with the A/C.  If it does, that means the speed used by A/C is dead.  Depending on the system, you might have to replace the blower, or simply run it at a different speed for A/C.

WARNING: Don't run the A/C with the wrong blower speed for long, as it could damage the system.

If it's an ECM motor, there may not be a way to select/change the speed manually. Or the speed selection might be adjustable using dip switches, or jumpers.