Cycle time is the time taken to produce a final part using different processes. Actually it is a max time taking process time.
Takt time is defined as the required time to produce one unit as per customer demand. It sets the rate of production to match the rate of customer demand
Takt time stabilize the system & control to overproduction.
When operators work as per takt time, their output increases.
Takt time helps the process designers to make line balancing.
More detail about Cycle time and Takt Time