Blower turns on when stat calls for heat but

Here is the ignition sequence for a spark ignited furnace. This will help you understand what is happening/not happening.

Thermostat calls for heat.

draft inducer motor (DIM) starts drawing draft through heat exchanger.

Pressure switch detects proper operation of DIM, and sends signal to board to begin ignition.
(it detects negative pressure)

Ignitor starts ignition attempt and gas valve sends gas to burners.

Upon ignition, flame sensor detects flame and sends signal to board stopping ignition sequence and continuing gas flow for heat mode operation until thermostat is satisfied.

During this sequence the blower motor is brought on, and by using a series of temperature limit switches, the system assures that the unit is operating properly.
(If the blower is not operating, or the airflow is inadequate, the heat exchanger will overheat and trip one or more temp limit switches.)

There is also a "rollout" switch that detects improper flame pattern "rolling out" into the controls compartment.

Now, to apply that to your situation.

If you have no spark, than the failure is happening before the gas is supposed to be flowing.

That usually means a failure at the D.I.M. Typical causes for that are a clogged flue pipe/vent, damaged/deteriorated fan blades, bad fan, or bad pressure switch.

If one of the temp limit switches or rollout switches have tripped, it will simply shut off the system.
(Some units will turn on the blower motor and shut down everything else.)

You can try resetting the unit by disconnecting power for 1 minute, and attempting a restart.

First look for any flashing LED lights. If you see any locate the wiring schematic (usually on the inside of the front panel) and look for the flash codes.

If the reset does not solve the problem, let me know what flash code you are getting, and/or let me know exactly what you see the furnace doing, in detail, after the restart.