The Harbor Freight motor is also an induction motor. "Compressor" refers to the rated duty cycle (e.g. rated for partial duty like 15 minutes for every hour, or continuous duty) and service factor (e.g. maximum peak HP that can be delivered for a short duration). Compressor motors should be continuous duty and usually have a 1.15 or 1.25 service factor to give 15% or 25% more horsepower for short durations.
A lathe motor needs to be the right RPM, continuous duty, 1.15 or higher service factor and sized so that the power extraction rarely exceeds the nameplate HP rating.
Some others posted good advice, verify the installation - is the motor turning the proper RPM, is RPM too high, is there any binding or resistance in the equipment being driven?
Harbor Freight would not be my choice for a replacement motor, reputable motor manufacturers will provide detailed specifications including the expected winding temperature under load which allow the end user to verify that the motor is operating within specification.
Ryan