In a watercooling loop, the water is moved by a mechanical pump; in a heatpipe simple convection moves the fluid. A water loop is a continuous 'loop' (thus the term) while the heatpipe's liquid follows the same path both to and from one end to the other.
In watercooling, the radiator is the primary heat exchange medium for the water; in a heatpipe the copper tube and the internal wicking surface is. In both cases the external, finned surface area helps improve dissipation with increased surface area.
Watercooling, the tubing (by comparison) is simply a transport container. In a heatpipe, the tubes' primary purpose is to contain the liquid + act as part of the cooling solution.
There are others, but these are ones off the top of my head.
Heatpipe has a major difference: it depends on the phase change of the liquid inside.
At the hot zone of the pipe the liquid evaporates and absorbs a ton heat through he latent heat of vaporization. The gas travels to the cold end of the pipe and condenses, releasing that heat energy into the copper pipe and then the fins. The liquid then returns to the hot area by using a capillary tube in the center of the pipe.
Water cooling is much more simplistic in that the water does not undergo phase change. It's advantage is the sheer amount of thermal transfer medium it is able to provide. You can only get so many heatpipes over a CPU and copper fins is usually too heavy. Also, heatpipes cannot be as long as this poses quite a challenge when setting up the evaporation/condensation cycle.
Otherwise, yes, both systems rely on a large surface area zone for which to dissipate heat over. Both systems require some form of airflow to remove the heat from the system. However, water cooling is capable of transferring that heat energy much farther away from the heat source than a heatpipe is and can provide a much greater surface area since it isn't limited to the space of the heat source.
It would be basically like having a pressure cooker. Water vaporizes at the heat source, rises and when it hits the top of the lid where it's cooler, it condenses and returns below to be reheated again.