You first need to understand that each of these are API's. In most API's, they are just a standard way for a developer to write something. On the backend, there is processing of these commands that actually does something.
Think of it as a calculator. There could be 100's of calculators on the market. But the API is the same. All of them take numbers, and operators (+-/*) and then there is that pesky = button. So when the programmer (user) enters the commands, it gives you the result. How the CPU inside the calculator calculates the equation could work really in any way, but to you, it's all the same because its the same API.
Now. When you think of it in this manner. Does OpenGL or DirectX have more overhead? It's really not a relevant question. It's just a way to translate programming commands to the GPU. DirectX was an API that was created that abstracts the underlying driver/manufacturer from the programming. So the programmer could write the program once, and have it work on any hardware. OpenGL is the same type of deal.
The difference here is implementation. With OpenGL, the video card manufacturers have to write the OpenGL to GPU implementation. So we can have wide variations in not only quality but also speed depending on driver versions and/or manufacturers or even GPU design.
Some of that is handled by Microsoft on the DirectX side. Such as memory management. My guess is that each video card driver (from the manufacturer) needs to write a memory management implementation for OpenGL. But on DirectX, Microsoft has done that within the DirectX API.
That could be a good or bad thing. DirectX maybe be more even in terms of performance/stability across vendors/implementation since it's doing more, but shortcuts could be made in drivers to improve the speed with OpenGL. But bad coding could make that implementation in the driver for OpenGL even worse than what Microsoft/DirectX has done, so there could be more overhead to OpenGL in some cases based on the driver, or it could be more efficient, you don't really know.
*Not really sure about all the internals, but this is how I understand it. It might not be very accurate.