Blower fan works w/ a/c “auto”, not with heat “auto”
Bad Blower Motor Coil
Furnaces typically have multispeed blower motors, and use different speeds for different modes.  If the coil for the speed the heat uses is dead, that would cause the problem you're seeing.
You can use an ohmmeter (or multimeter set to measure resistance) to check if the motor coils are good.  You'll have to measure the resistance of each coil (while isolated from the rest of the system), and compare the results to the manufacturer's listed values.
Bad Control Board
There are a few things that could go wrong on the control board, that could lead to the issue you're seeing. The first thing to do, is to determine if power is ever getting sent to the motor at all.

THIS IS DANGEROUS
Since you'll be working on a live HVAC system, this testing can be dangerous.  If you're not comfortable, please contact a local licensed
HVAC technician.

Set your multimeter to measure AC voltage, and connect the common lead to the neutral terminal on the system.  Hold the door lockout switch in, so that the system will try to fire. Once the burners ignite, listen closely to hear any relays closing.
Most furnaces use timers to delay the blower, to allow the heat exchanger to heat up. Once the preset delay has elapsed, a contact should close and send power to the HEAT terminal on the control board.
Put the other lead of your multimeter on the HEAT terminal, and see if you get a voltage reading. If you do, then the problem is with the motor or the wiring to the motor. Check all connections, to make sure they are clean and secure.  If you don't read voltage, you'll have to either troubleshoot the control board, or simply replace it.