Well, that depends. For gaming, DirectX is generally preferred. At least in the past NVIDIA cards performed better in OpenGL applications, while ATi was better in DirectX, I don't know what the current situation is. Unlike DirectX, OpenGL is a open-source graphics library and hence is used for graphics-related stuff/games on Linux (and other non-Windows platforms). I don't know too much about technical/"under-the-hood" differences, though.