It´s all about preference, if you were to browse through one or more, there come in the thousands, graphics programming forums or sites you will quickly see hundreds of posts arguing which API is better. It´s the standard DirectX, Direct2d and Direct3d, which were merged into one when Directx8.0 was released - DirectGraphics, versus Opengl.
I get the impression that a lot of people seem to think that OGL is fading away, leaving more room for DirectX. You can accomplish more with DirectX, since it´s a wide API, besides graphics(DirectGraphics) it covers input(DirectInput), audio(DirectSound, DirectSound3d), network/internet play(DirectPlay). Games often use OGL for graphics and Directx for audio and input, so it that sense Directx is superior to OGL, it covers more terrirory.
But when the rest is peeled off, and a naked DirectGraphics tries to knock down OGL it will not succed, neither will OpenGl. OpenGl is a graphics only API, and despite that Directx is a much more mature API, in the latests years it has gone from version 1.00 to version 8.1a, soon to be version 9.00, while OGL remains at version 1.x, OGL version 2.0 is being developed, OGL can achieve everything that DirectGraphics can. When the OGL standard has halted, it´s not probaly not the right word, but it has evolved slower, and DirectX has accelerated in development, OGL has been updated and modernised through extensions. So to sum it up...
DirectGraphics can achieve everything that OGL can, and what OGL can not achive naturally, compared to DirectGraphics, it does through extensions. DirectX is also platform specific, Windows, where OGL is platform independant. Other then these major API´s there´s only SDL, Simple Direct Media Layer, left which to my knowledge has yet to be used in a Zilmar spec Nintendo 64 emu, regardless of form - audio, input, graphics.
It should also be mentioned that DirectX is mainly used for entertainment, gaming, where OGL is geared towards more professional applications, military simulators uses OGL as the graphics API.