OpenGL is cross platform, fast and creates new technologies yet alone following them. Unfortunately doesn't have a good driver support especially by ATI/AMD.
DirectX is Windows only (and yes, Xbox), not as fast as OpenGL, DirectX 11 has yet started to follow latest technologies, Microsoft monopoly. Has a good driver support.
At the end of the day, you are going to have to research and decide what tool is the best for the goals you have in mind. Other than that, pick the one you feel you are more comfortable using.
(This post was last modified: 07-07-2011, 02:28 AM by Apjjm.)