What is the difference between manual, automatic transmatic and hydrostatic drives in a ride-on mower or tractor?

Manual Drive Systems: As the name suggests, the manual drive is a traditional style gear lever (or stick-shift) system. This will provide a number of gears, usually including a reverse, that offer you a range of speeds, depending on the number of gears provided. 

Automatic Drive Systems: These belt and pulley variable drive systems which, like hydrostatic drives, eliminate the need for manual gear changes. The operator only uses the drive pedal to accelerate and needs to use a separate brake pedal to slow down and stop. 

Transmatic Drive Systems: These systems use drive-belts and tensioners to control the power and therefore the speed, and transfer it to a transaxle. An operator presses the clutch and manually shifts into the chosen speed and then releases the pedal. The difference from manual gears is that the operator does not need to accelerate through the gears as if in a car. He/she would simply choose the speed and then work at that speed. 

Hydrostatic Drive Systems: This totally gear-free system uses hydraulic pumps to control the velocity and direction of the ride-on. Easy to use, the operator simply presses the pedal and the further down, or harder, it is pressed, thew faster the ride-on or tractor moves. This makes it very easy to manage with infinite and detailed speed control and a full stop if the foot is taken off. 

Mountfield Petrol Tractor with Pedal Control Hydrostatic Drive