Historically, OpenGL has always been *easier* to code for, faster code execution, and generally able to compete based on the fact it's a graphics library and *NOT* a 3D Gaming API. That's not what it's for at all, it's a Graphics Library.
DirectX, on the other hand, has been more interested in providing a one-size-fits-all solution, doing user-input, networking, sound etc, all of which is rapidly becoming deprecated in the eyes of Microsoft.
As a 30,000 foot view of the two projects would imply, OpenGL defines a series of methods of doing things, with common physical/mathematical operations being specified as functions to make it both easier and faster to do things, while DirectX focuses on defining a new specification and forcing the hardware manufacturers to "keep up".
DirectX is the monopoly, beating manufacturers with a stick and screaming at customers going "I'm better!" while OpenGL has just got on with the job. The fact is, these days, there isn't much between them, and it comes down to what your use case is. If you want something that is going to be used in a gaming environment, you tend to lean towards DirectX. If you want something that is going to be used for a simulation environment, you tend to lean towards OpenGL. YMMV.