A Squirrel Cage Induction Motor (SCIM) only has a cage rotor winding, whereas a synchronous machine has both a cage (damper) and a wound DC field winding as shown in figure below.
Synchronous Motor with Damper Winding:
Squirrel Cage Induction Motor:
In both cases, and when using only the cage winding, both motors will run close to synchronous speed just like a normal SCIM. However, when the synchronous motor field winding is supplied with DC, it will lock into synchronism and its cage or damper winding will carry zero current as the relative speed between the damper winding and rotating flux is zero. Under load disturbances the damper winding will have current induced in it, develop additional torque, and thus help to damp out transient disturbances.
In smaller synchronous motors the damper winding is used to start the motor just like a normal SCIM and when close to synchronous speed the DC field is applied and the rotor locks into and runs at synchronous speed.