Hydraulic systems and compressed air systems are usually regarded as separate things. Maybe the distinction is a bit arbitrary and the former can be regarded as a subset of the latter but there you go. edit: Google says hydraulic transmits power through fluid, compressed air through air/gas.
Pneumatic vs hydraulic is about compressible vs incompressible fluids. (Naturally, hydraulic oil is also a much better lubricator and an exceedingly superior sealant than oiled air).