What is the difference between driving a MOSFET gate and an IGBT gate?
Sometimes...

Assuming the point of interest is power MOSFETS and not small signal MOSFETS and silicon (as oppose to SiC, GaN) 

The first characteristic to check is the output voltage. For power devices they should be 0V to 12-15V (acpl-312T) to cater for gate thresholds around 4V (as well as being able to drive to -15V if miller turn-on is a concern). As such a MOSFET driver driving an IGBT &amp; equally an IGBT driver driving a MOSFET should be fine.

The next characteristic is peak current. IGBT's will have significantly larger gate capacitance and as such will require higher peak currents to ensure the device saturates as quick as possible. The converse of this is MOSFET's can be switched faster and as such the rms current demand to drive a MOSFET might be higher. 

Higher current or higher switching frequency affects the driver power capability.