I have had the same set up for years. When I turn to "cool", the A/C unit kicks on, but the blower does not. The only way I can get it to work is by turning it to "ON" instead of "Auto".
In years past, it has always worked on "auto"... What could have happened?

Probably the fan control,, when in heat mode, does the fan kick on and off like it should? Not knowing the brand or model mumber of you furnace/airhandler I couldn`t say for sure, do you have a control board? Some fan controls are built into the board

The heat mode works just as it should. My furnace/blower is a Frasher-johnston model G1fao... Its about 3 years old.

It sound like you have a bad control board. When the thermostat calls for cooling it sends 24v to the Y and G circuit at the same time, telling the system to turn the blower on high speed. You may need to check for 24v at the air handler on the G circuit to rule out a bad thermostat.

Most likely a thermostat??

Not most likely. Just a chance. That's the reason for testing it so you don't make a costly mistake.