Or, as I learned in tech school:

Pneumatic systems use compressible fluids. Hydraulic systems use non-compressible fluids. Pneumatic systems can use their medium to buffer changes in fluid movement. Hydraulic systems cannot without having additional hardware installed.

Hybrid Systems and Non-Homogenous Phase Cross-Terms

Neither Hydraulics nor Pneumatics are, however, the most common
systems except where humans exert effort to cause them to be.
The nearly ubiquitous system is the hybrid system. That is,
systems which must deal with both liquid and gas phase fluids in
their conduits at the same time. Usually, these systems have a
dominant phase of medium. In other words, most systems are
pneumatic systems which must tolerate some liquid in them or
hydraulic systems which must tolerate some gas in them. Biological
systems are a good example. Respiratory systems are primarily
pneumatic but have sigificant operating characteristics due to a
small but not insignificant amount of liquid(mucus) in them.
Digestive systems are primarily hydraulic with a small but not
insignficant amount of gas(flatulence) in them. Home water supply
and waste water systems are similar. Supply systems are primarily
hydraulic with tolerance for air present in the system(bubbles).
Waste water systems are even stranger since they are hydraulic
systems with gravity rather than pump powered fluids and large,
significant volumes of gas(air) in them.

Often, significant operating mechanisms in these hybrid systems
come from the medium with the opposite phase as the primary working
fluid phase of the system. Lubrication is often the operating
mechanism which comes from material in the system which does not
match the phase of the primary operating fluid. Pneumatics often
use liquid phase lubrication(oil). Hydraulics often use gas phase
accumulators. Even in human designed systems, hybrid phase is
commonly used to achieve lubrication such as the oiler mister in
many shop air systems. Shop air is interesting since the traditional
filter, drier, oiler sequence of mechanisms is manipulating non-primary
phase elements in the conduit. "Filter" removes solid phase contaminants.
"Drier" removes water, a liquid phase contaminant. Then "oiler" injects
a required liquid phase lubricant into the predominantly gas phase
system.

Human created hydraulic systems can become faulty if significant amounts
of gas phase fluids are present the conduits. The fluid for which
hydraulic systems are named, water, is frequently rejected as the working
fluid for the system. Water is a wonderful hydraulic fluid with two
significant exceptions. The first is, that while water can behave as
a lubricant, it is also such a good solvent that it also tends to remove
lubricants designed to stick to the system's moving parts. Without proper
lubrication, friction generates heat, and the heat can evaporate the water.
The vapor phase of water is compressible, unlike its parent liquid fluid.
That compressibility disrupts the good hydraulic fluid characteristics
right where the system critically depends upon them. Therefore, most high
force, high power, hydraulic systems shun the very same eponymous fluid
that gives these systems their appellation. That is, hydra means water
but water usually cannot work well in extreme hydraulic systems with local
temperatures high enough to evaporate it.
_______________
thewildotter

Greek prefix etymology:

HYDRA = water, wet.

PNEUMA = air, breath.