Well, You should have said : D3d vs OpenGL. Since Direct X has other non 3D gfx related items like Direct sound, Direct play and Direct music.
For me I prefer D3D over OpenGL since it requires less coding experience and still gives nice results compred to OpenGL. But if you really let me choose the perfect API it would be Glide which made it's job perfectly and gave higher frame rates for voodoo cards. And when it was first introduced neither D3D nor OpenGL could compete it.
I'd say D3D too, the way things stands now D3D have many more functions, and you'll have to write a lot more yourself with openGL, unless you use something like openscene graph, which is mostly a set of functions for openGL already included in D3D.
The only reason to use openGL ( which could be a big one though ) is if you want to devellop for the PS3 in the future.
LOL, I'm talking from a developers perspective here, I guess from a game perspective is intressing to though, Doom 3 is the only game I've seen that openGL has going for it, and well, to be perfectly honest that's just because Carmack wrote most of the advanved functions for that game himself
OpenGL has fairly simple and efficient portability & is easily modified and changed around by the coder, but function wise, it's limited in comparison to the ever growing D3D. OpenGL, as an API, doesn't determine how good games built around it will be.
CloudVII: You're thinking of the DirectX set as a whole.. this is a D3D (DirectX's 3D API) vs OGL.
Doom III does prove what can be done though, so comparing is pretty useless.
The impression I get at the oment is that it is easier to do funky looking stuff in D3D, but OpenGL is more flexible. However, I'm looking at this from the end users perspective given that I'm not a programmer.