A few years ago OpenGL was definetely in the lead, we even wrote an open letter to microsoft trying to get installed as standard and for them to support it, now hardly any of the game developers care, DirectX changes faster, they sorted out nearly all the API mess with DX8, DX( is well embeded into the world.
Though NVIDIA etc usually exposes their hardware first in OpenGL, big suprise NVIDIA is made up of primarily ex-SGI OpenGL people.
OpenGL is better for cross platform obviosully, but again for current games no one ( developers) really cares about that, the other markets aren't really viable, and for the console development its not as good an idea to abstract the API anymore, most bang for the buck is writing as close to hardware as possible, PS2, shaders etc,
OpenGL 2.0 and DX10 is when it'll start to get interesting again.
As for one looking better, its unlikely that one API when used correctly will make something look better, typically its the implementation by the developer, the primary reason for this is that the OEMS make the drivers for their own cards, and they want them to look the best they can on both APIs, they have nothing to gain by not making them the best they can be, and these guys are fanatical about getting the best out of their cards.
Its possible to get the same image quality out of either, as its that hardware that matters, and its the same hardware that does the actual rendering no matter which API is used. just not everyone does.
Sorry to ramble
