In particular, relating to -Speed of Response-Force Available-Supply of Operating Fluid-Weight of system
Hydraulic systems are actuated by the displacement of a fluid, usually oil. Pneumatic systems are actuated by the displacement of a gas, usually air. Oil is not compressible and air is; therefore, hydraulic systems have a much shorter response time (almost instantaneous) with greater force transmitted through the system. Oil weighs more per volume than any gas, and the components must be more robust to manage the greater pressures, thus hydraulic systems tend to be heavier than pneumatics. Hydraulic systems rely on a consistent supply of fluid in the closed system. To replace a fluid loss from, say, a leaky gasket or fitting is often more troublesome than starting a pump to maintain pressure in a pneumatic system by drawing in more ambient air. One system is not necessarily better than the other. It depends on what you want done within which limits.
The systems are quite similar in a lot of ways. Hydraulics can give you higher forces than pneumatics. In a hydraulic system the pressure is generated by a positive displacement pump, in pneumatics it is generated by a compressor. Air is a lot more compressible than hydraulic fluid, so there is another big difference. Both systems use control valves that work in more or less the same way. Hydraulic lines are designed for laminar flow (I think it has something to do with air mixing into the oil, but I cant remember if this is right). I am not certain about whether this is any sort of problem in pneumatic lines (I mean laminar vs turbulent flow, I don't think air mixing with air should be much of a problem lol). In a hydraulic system the fluid gets returned to the reservoir when a piston retracts. In a pneumatic system the air get vented to atmosphere through the control valve when pressure is not needed anymore. Or in the case of a motor the air just goes through and gets vented immediately.
Hydraulic systems utilize fluid to transmit force. Examples are brake systems, landing gear struts and (in more complex aircraft) flight control or landing gear actuator systems. Pneumatic systems utilize gas (usually air) to transmit force. Examples are gyroscopic instrument air systems and (in more complex aircraft) deice boot inflation systems. Since gasses are compressible, and fluids are not, hydraulic systems have a greater speed of response and transmit higher forces. Pneumatic systems are generally supplied by atmospheric air (pumped), or engine bleed air from the compressor sections of turbine engines. Hydraulic systems are supplied by on-board reservoirs and/or accumulators. Pumps are used to amplify control forces input by the pilot when higher forces are required to move an actuator. Pneumatic systems are generally lighter than hydraulic systems, due to the absence of fluid. If you are a student pilot or mechanic, please take time to go back and study these topics in your coursework. Go to an aircraft and touch these things. Look in the systems description section of any moderately complex aircraft POH or AFM. This kind of systems knowlege is very important to your confidence and safety when you are operating or maintaining an aircraft, especially when things go wrong. You really really need to be confident that you know how this stuff works.