In a synchronous motor, Rotor and stator are excited using different excitation sources. Furthermore, rotor and stator are always synchronized and move at both moves at the same synchronous speed. This synchronization is achieved with the help of separate excitation sources for rotor and stator (as explained above).
However, in the case of induction motor, as the name suggests works on the induction of current in rotor due to changing flux in the stator. When Voltage is applied to the stator of the induction motor, the current will flow through the winding, resulting in an induced current in the rotor due to changing flux in the stator. Due to this current flow in the rotor, a lagging flux will also produce and finally, the rotor will rotate due to the torque caused by these fluxes. As rotor flux is dependent on stator flux, so it will always lag with the stator, and hence rotor speed will always be less than stator speed.
Hence, the induction motor is called an asynchronous motor, because there is no separate excitation source available for the rotor. Rotor movement is dependent on the induction of current in it. Rotor always rotates at less speed than synchronous speed, as it is always lagging from the stator.
In short, stator and rotor do not rotate at the same speed, resulting in no synchronization between rotor and stator. Thus, the induction motor is an asynchronous motor.