What's the difference between a flight dispatcher and an air traffic controller?

Flight dispatchers work for airlines, planning flights. They calculate altitude to be flown, fuel burn, weight and balance of the aircraft, etc. They plan the flight to be the most efficient as possible. Dispatchers require a license.

Air Traffic controllers ensure the safe movement of the aircraft from the start of the flight, until it is safe on the ground.

A flight dispatcher works for a company, ie United, Delta. They are the ones that work out the schedule for the particular aircraft to a destination.
An Air Traffic controller works in the control tower and separates aircraft in their zone for safety. They are Gov employees. You have approach, departure, ground and air traffic control. Those are the ones you hear the most in flight.

The Air Traffic Controller ATC are mostly government employees and here in US employed by FAA if I am not mistaken. They are positioned at ATC control towers at airports to regulate movement of aircrafts giving out engine start up, taxi and take off clearances where as other ATC s positioned at Approach control separate and regulate air traffic flow in the "approach area " of the airports for smooth and maximizing the traffic flow into and out of the airport.

Flight Dispatchers are Air line operators employees who are trained for safe despatch of the flights from a given airport, taking care of weather at departure, enroute and destination airports, fuel requirements of particular flight in accordance with fuel policies of the airline, timely filing the flight plan routing with the ATC and keeping track of actual departure and arrival times of the airline flights from the airport.

These two employments are different in many ways but both play a specific roles in the safe operatin of the flights.

To begin, I've been a licensed practicing Part 121 dispatcher for almost 30 years. This is the distinction; the dispatcher shares operational control with the captain for the planning and operation of the flight. The act controller sole responsibility is separation of aircraft. As when I addressed a group of ATC controllers, "I told them, that they did not have operational control." They as controllers did not have the authority to change the flight's route of flight, or that flight's destination or terminating that flight. Only the captain and the dispatcher have that authority. You'd have thought a stake had been driven through Dracula's heart by looking at the faces of those controllers. The FARs specify, that only the flight's Captain or Dispatcher can change the route, destination or terminate a flight. Only the Captain & Dispatcher have the knowledge regarding fuel planning & performance. During 9/11, ATC was directing flights to land at the nearest airport without regard to runway weight bearing capacity or runway length. The FAA at times likes to impose itself into the operational control and the decision making of the daily operations of the airlines. The FAA & VIPs have no qualms what so ever to inconvenience the traveling public when safety is not being compromised.