Hydraulic winches are slow and they don't have the heating issue if used for long period of time like an electric one, but as you pointed out the motor has to be running to use it, which is really the case for electric one because they can draw a battery down quick if your not careful. Also in general you can get a electric one easier and cheaper then a hydraulic. The other thing is that the only hydraulic one that I know of was made by Mile Marker and from what I've been told they stopped making them, so you will only find them used.
There is another option with electric's winches and that is going to a worm gear drive system over the standard plantatry drive, but the catch with the worm gear drive models is that they start off arounn $1500 and up.
Hopes this helps Jason.