Independent game company Wolfire write why you should use OpenGL and not DirectX. The article goes over a brief history and the standard and Microsoft's tactics with DirectX, and what this really means for developers.

It's common knowledge that OpenGL has faster draw calls than DirectX (see NVIDIA presentations like this one if you don't want to take my word for it), and it has first access to new GPU features via vendor extensions. OpenGL gives you direct access to all new graphics features on all platforms, while DirectX only provides occasional snapshots of them on their newest versions of Windows. The tesselation technology that Microsoft is heavily promoting for DirectX 11 has been an OpenGL extension for three years. It has even been possible for years before that, using fast instancing and vertex-texture-fetch. I don't know what new technologies will be exposed in the next couple years, I know they will be available first in OpenGL.

Microsoft has worked hard on DirectX 10 and 11, and they're now about as fast as OpenGL, and support almost as many features. However, there's one big problem: they don't work on Windows XP! Half of PC gamers still use XP, so using DirectX 10 or 11 is not really a viable option. If you really care about having the best possible graphics, and delivering them to as many gamers as possible, there's no choice but OpenGL.