The cooling test sequence for this board, says to jumper Y & G to R. Make sure the fan comes on, and spins up to the proper speed. Then remove the jumpers, and check the fan off delay.
If the blower does not come on, verify that the COOL terminal is energized. Use a voltmeter between COOL and neutral, and verify 120 volts (with jumpers in place).
If the COOL terminal is energized, check the motor and verify all speeds are working properly.
If the COOL terminal is not energized, the board my be bad.
According to the schematic you've provided, you have a multispeed blower (which is quite common). The speed at which the motor runs, is determined by which wire is energized. Energizing the red wire; for example, will run the motor at low speed. Energizing the black wire, runs the motor at high speed.
Before you disconnected the board, one of the motor wires was connected to the HEAT terminal on the board, and one was connected to the COOL terminal. Looking at the ladder diagram, it looks like the blue (MED) wire was connected to the HEAT terminal, and the black (HI) wire is connected to the COOL terminal.
If you've run the tests described above, and found that the COOL terminal is energized when the jumpers are in place. You can try connecting a different speed wire to the COOL terminal. If the motor spins up with the other wire connected, it means that the original speed is dead or something's wrong between the board and the motor.
It looks like the LO speed red wire is attached to the M1 terminal on the board. I'd swap that to the COOL terminal (just temporarily for the test), and see if the motor spins up when the jumpers are in place.
WARNING: DO NOT run the A/C for long with the motor at a lower speed..
If you want anything close to an accurate answer. You're going to have to tell us a bit more about the wiring between the thermostat and the furnace, or include some photos of the wiring. And include make, and model numbers for all the equipment involved.
Most furnaces manage the blower themselves during a heat call, i.e. the thermostat simply energizes the W wire. Some also manage it during a cool call, but not all do. So when calling for cooling, the thermostat usually energizes both Y for cooling, and G to turn on the blower.
Since you say the blower doesn't come on for cool calls, or when the fan switch is moved to the ON position. That points to a problem with the "fan circuit". Between the thermostat and the furnace, there should be a wire (usually green) connected to the G terminals of both devices. When this wire is energized, it tells the furnace to start the blower. Some furnaces have an on/off delay, so the blower may not come on immediately. On some units, this delay is adjustable.
Since you haven't told us the make and model of any of the equipment, there's no way to be specific about what happens in your furnace. Some furnace control boards have terminals for HEAT, COOL, and FAN. While others may only have HEAT, and/or FAN. The motor speed wires are connected to these terminals, to allow the furnace to run the blower in different speeds depending on the situation.
The first thing I would do, is test for voltage between the grounded ("neutral") and the COOL terminal on the board, during a cool call. If you measure about 120 volts (line voltage), that means the board is energizing the terminal to turn on the blower. This means there's a problem between the board and the motor, or with the motor itself. If the terminal is not energized, then you'll have to start moving "back" through the circuit.
If the furnace is signalling for the blower to turn on, but the blower is not starting. You could try connecting a different speed wire to the terminal. Since the blower works for heat, you might try putting the wire that's on the HEAT terminal on the COOL terminal. If the blower fires up, you know it's a problem with the motor. WARNING: Don't run the system this way for long, you're just checking to see if the blower starts.
If the furnace is not signalling for the blower to turn on (no power to COOL terminal), then you'll want to start by making sure the thermostat call is reaching the furnace. Disconnect the wire connected to the Y terminal, and set the thermostat so it's calling for cool. Measure voltage between C (or the common side of the transformer), and the wire that was connected to Y. You should get about 24 volts AC. If you don't there's a problem with the thermostat, or the wire between the thermostat and furnace. If you do, then the problem likely lies within the control board.
Without more details about the system, it's impossible to offer any more specific advice. It's possible that there's an open limit, which is preventing the blower from turning on. However, without a schematic, there's no way to know if there are any limits that might be involved.
Best Answer
The contacts in the thermostat for the "fan on" are different than for the "fan auto". If you know what type of thermostat you have you should be able to find a wiring diagram and determine which wires to short out on the mounting block to simulate "fan on". In my thermostat, it's red and green but you'd have to verify that with yours. You could also remove the thermostat with the "fan on" in the "on" position and test for continuity between the contacts.