The big disadvantage here is the loss of precision due to the high compressibility of gas compared to liquid. Because gases are highly compressible, they provide a buffer to changes in pressure commanded by the operator to move the piston in the cylinder. That poses two problems; first, it means that the pneumatic cylinder doesn't respond instantly to pressure differentials, because the differential must first overcome the cylinder gasket's static friction. Second, it means that the movement of the cylinder is more easily opposed as long as whatever force opposes the gas pressure can overcome said pressure without causing whatever the pneumatic system is controlling to fail.
To overcome these shortcomings, most pneumatic systems run at very high pressures, so that the pressure differential between the two halves of the cylinder readily overcomes static friction and any other opposing forces. However, that creates another precision problem; high-pressure pneumatic cylinders are essentially two-state systems; the piston or actuator is typically at one or the other of its extremes of movement, and transitions between them very quickly as gas pressure is applied to one side or other of the cylinder.
None of these behaviors are desirable for aircraft controls; instructors labor daily to teach their students not to ham-fist the controls, instead using a bit of finesse to get the plane to do what they want in a smooth, controlled fashion. Why then, would you want to undo all that finesse with a control system that can only move the surface to the extremes of its travel?
Hydraulics, by contrast, allow a much higher degree of finesse. Because liquids don't readily change density, the pressure changes within a hydraulic cylinder require much more force to oppose, but by the same token, as the volume changes the pressure on the side being supplied with fluid decreases rapidly. This allows a hydraulic cylinder to be positioned much more accurately, regardless of any external forces acting on the system. The disadvantage is hauling a fairly heavy liquid up into the air, and having only limited capacity to replace it if any of it leaks.
Electrical actuators are a common solution to that disadvantage, especiually in light aircraft. Electrical actuators use an electric motor or servo to provide the mechanical action. These actuators can be controlled with a high degree of precision, and their "supply system" is just an electrical circuit, no heavy and complex hydraulic lines and cylinders. Their disadvantages are a tradeoff between speed of movement and maximum applied force while moving; you can either make an actuator that moves very quickly, or an actuator that will move no matter how much force is opposing the movement, but you really can't do both. They're still useful in light aircraft to control flaps (with a cable system used for the main surfaces), because they allow for precise amounts of extension or retraction, and don't have to instantly respond to input like the primary control surfaces do.
There is something on the horizon that could make pneumatics feasible for aircraft. Hydraulics systems were recently improved with the development of the electrohydraulic servo valve. This system uses a variable electrical potential (voltage) to move a hydraulic cylinder by a prescribed amount proportional to the voltage applied. Pure electrical servos have been around for decades, but the maximum amount of force available from a servo is insufficient for large airliners, while for smaller aircraft the servo motor's relatively high weight compared to simple cable controls is a disadvantage. The electrohydraulic servovalve concept is used in newer large aircraft to replace pure hydraulic or cable/hydraulic hybrid control systems, because the hydraulic system can now be controlled by an electrical circuit instead of hydraulic lines or tensioned cables coupled to the control column. This allows for "fly-by-wire" aircraft such as most Airbus airliners as well as most fighter jet designs of the last 40 years.
A similar concept is under development for pneumatics, allowing the precise placement of an actuator using pressurized gas in response to an electrical voltage. This would provide all the advantages of an electrohydraulic system, with considerably lighter weight and faster response, but still having the disadvantage that a significant opposing force could prevent movement of the actuator especially as it approaches the desired position. Whether that will be an issue in a large aircraft remains to be seen, and the weight savings of losing the hydraulic fluid might not be worth it, but if the tradeoff is acceptable, it would further increase range or payload of the next generation of passenger aircraft, with the added safety/reliability feature of being able to compensate for a slow leak in a pneumatic system by simply adding more air with a compressor pump.
