No, they are not the same thing.
Hydraulic drive uses a gear type pump that moves a predetermined amount of oil every full revolution of its input shaft. As an example, a gear pump with an internal displacement of ONE cubic inch will transfer one cubic inch of oil from the intake side to the discharge side each revolution. If you spin that pump at 3600 RPM, then the pump will move 3600 cubic inches of oil every minute.
Since their are 231 cubic inches in a US gallon, that means a one cubic inch pump will move 15.58 gallons per minute as a constant flow. These pumps are known as fixed displacement pumps. There is no way to change the amount of output. They either pump that much oil or they blow apart from the pressure they can create if someone attempts to block the output 100 percent and not provide an emergency escape route.
Therefore, the oil in a Case tractor is in constant circulation, moving from the pump to control valves that can re-direct the oil flow as needed before continuing to an oil cooler and then the oil reservoir. At that point, the oil then heads to the pump for another trip through the system.
A hydrostatic pump is totally different in design. It is known as a "variable displacement" pump because a person can push/pull a lever or step on foot pedals and change the amount of oil pumped. When you jump on a tractor with a hydro pump and start the engine, the internals of that pump are spinning but they are not pumping any oil until you move the control lever or step on the foot pedal. When you do either of those things, a lever on the outside of the hydro pump rotates a shaft that is connected to a "swash plate" inside the pump and causes it to "tilt" away from being square with the body of the pump.
That plate can be tilted either way from center. Tilting that plate cause a series of pistons to move back and forth. Those pistons reside inside holes in a rotating drum. Visualize an old Colt 45 handgun. It has a drum with 6 holes in it to hold the bullets. That drum is being spun by the engine and the bullets are the pistons which move in and out of the holes in the drum thanks to the swash plate being tilted just like the throttle and choke plates in a carb get tilted. The more you tilt the swash plate, the further the pistons travel up and down inside the rotating barrel and the greater the amount of oil is pumped.
That is what makes a hydro pump a variable output pump.
A hydro pump has a port on either side of it so that high pressure hydraulic hoses can be connected and those lead to the ports on a hydraulic motor. If you tilt the swash plate one way, the oil will be forced out the left port of the pump and oil returning from the motor will enter the right port of the pump. If you tilt the swash plate the other way, then the oil direction is reversed. This allows the operator to instantly change the direction of the tractor as well as control the ground speed.
This is called a "closed loop system" and in order for it to stay cool and have a constant supply of oil, a 2nd pump is needed. That pump is housed within the hydro pump's case. It is called a "charge pump" and it is a small, fixed displacement gear pump. It is there to circulate a small amount of oil from the hydro pump to the reservoir and back again. Doing so allows the oil to be cooled because high heat is a mortal enemy of hydraulic systems.
The problem with a hydro pump is that it can only pump oil to the motor that makes the tractor's wheels turn and that only happens whenever the operator wants the tractor to move. The rest of the time, the hydro pump just sits there and spins without moving any oil at all. However, the charge pump constantly moves oil through the hydro pump in order to keep it cool. We can tap into the flow of the charge pump and use that flow for things such as power steering or a hydraulic cylinder on a power-angle utility blade.
As good as that seems, the charge pumps only put out a relatively small amount of oil and at a modest pressure. We are talking about 3 to 5 GPM @ 800 to 1000 PSI. Now that is fine for cylinders and you can operate a FEL quite nicely with a pump like that. But you cannot run hydraulic motors of any size with that low of a flow and that low of a pressure.
The hydraulic system on a Case or Ingersoll tractor puts out as much as 9 1/2 GPM @ 2200 PSI. This high flow system is what allows the use of a reversible direction hydraulic rotary tiller or a 4 foot rough cut field mower. Other motor powered attachments are a chipper/shredder, 3 pt hitch finishing mower and a suction fan that pulls debris from the mower deck and deposits it into a trailer or mesh bags.
Both systems have their merits and their short comings. The perfect tractor would employ both pumps but cost is always a consideration when you are trying to sell a product.
Skid steer loaders use both types of pumps. They have one hydrostatic pump to power the left wheels and another to power the right wheels. The loader functions are handled by a fixed displacement gear pump. ZTR's use similar systems.