Etymologically speaking in a synchronous motor, the rotors rotate at a speed similar to the speed of a magnetic field, or stator field, while asynchronous motors have the rotor running at a slightly slower speed than that of the induced magnetic field.
asynchronous
Asynchronous motor (above) vs Synchronous (below)
synchronous

Synchronous motor Asynchronous motor (induction)
More power at the same speed No brushes, i use the electromagnetic induction faraday. The rotor is made by laminated pieces, to dissipate eddy currents caused by changes in B and also hysteresis loss in ferromagnetic materials
ω=kf, so the rotational speed is always k times the input frequency Cheap and made by few parts, so reliable as well
Windings on both stator and rotor Presence of slip: the 2 rotating field are asynchronous
Easy to control and smooth functioning Squirrel cage into the rotor, connected only in the ends. No rotor windings
More complex in building Current in stator creates a B in stator who induces I into rotor and the consequently B over there. The motion is the consequence of the interaction of the 2 magnetic fields
More unitary power (pay attention to the heat dissipation) and performance η The cost is 2-3 times higher

Moreover

i would like to share some of experiences of motor lamination manufacturing: the process of the stator and rotor lamination process as below: