A Free vortex requires no external torque. It occurs when the water drains out of a tank or container. In a free vortex, the velocity is lowest in the outer periphery and it progressively increases towards the centre. Thus, the velocity profile is inversely proportional to the radius of the vortex.
A Forced vortex is formed when an external torque is applied. An example would be fluid rotated by a pump impeller. In a forced vortex all the particles are at the same velocity.