As a hydraulics layman thinking about hydraulic systems, it seems that the important factor is to have a liquid that doesn't compress much or at all. Doesn't water meet this requirement, and what other properties should the liquid have (if any) that water doesn't?
Water meets the low compressibility requirement, but there are many other considerations in the design of a hydraulic system:


Boiling point/vapor pressure: If the system warms up during operation, the fluid may boil, which results in high compressibility and thus decreased effectiveness of the hydraulic system. Hydraulic fluid has a higher boiling point than water to help combat this. Related to this is the concept of vapor pressure. Hydraulic systems often involve small orifices, which can cause cavitation (localized boiling). This cavitation has the same effects as boiling and can cause pitting damage to the components near the cavitated region. Hydraulic fluid has a lower vapor pressure which helps here.
Freezing point: It would not be a good thing if your car's brake lines froze every time it got cold outside. Most hydraulic fluids have much lower freezing points to prevent this from happening under normal circumstances.
Oxidation/corrosion: Water, being an electrolyte, will cause rust inside the lines as soon as air inevitably leaks into the system or the system isn't bled properly. Water will also exacerbate galvanic corrosion when dissimilar metals are used in the system.
Lubrication: Hydraulic components use seals and often involve sliding interfaces (cylinders and spools, for examples). Using an oil as the fluid means the working fluid can also function as a lubricant.
Organic growth: If perfectly distilled water and a closed system could be guaranteed, this would be a non-issue. But in practice, this is never the case. Oil-based hydraulic fluids are much less conducive to organic growth than water.


Water is used in some systems where other considerations trump these (for example, some food-grade applications), but for a wide variety of applications, oil-based hydraulic fluids are the better choice because of the design considerations above.
