Both work fine, D3D's much nicer, usually your platform dictates the choice. That's pretty much it. At the end of the day, learning modern GL (ie 4.0 and ES 2.0) prepares you for most of the non-Windows platforms you're likely to see, and that makes it a compelling choice even though the API itself is fairly shoddy.

The API is largely irrelevant - if you learn one of (modern) OpenGL or Direct X, you should be able to pick up the gist of the other in a couple of days.

Provided, of course, that you actually learned the theory at the same time

In my own experience OpenGL is alot easier to learn for a beginner. DirectX is a little more diffucult. But like switftcoder said learn one and you should pick the other up pretty quickly.

I forgot to mention, also, that both APIs are essentially a minor part of your code these days. Once you've got the concepts down, it's more just a question of wiring up the API to do things the way you need. It seems like a really big deal as a beginner to be coding all this D3D or OGL stuff, but I can honestly say it's probably the most boring part of my graphics coding work. The API is just a machine I have to configure so that my CPU code and my GPU code talk to each other properly.

I can't speak for every game developer, but we work with whatever API is available on the hardware. Windows is the only place we have a choice, and we happen to use D3D11. I'm sure other studios use OpenGL on Windows. Personally you'd have to pay me a *lot* of money to choose GL over D3D11 on Windows, but that's just me. Most people probably wouldn't care that much.

In my own experience OpenGL is alot easier to learn for a beginner. DirectX is a little more diffucult. But like switftcoder said learn one and you should pick the other up pretty quickly.

I disagree. Even comparing oldschool GL to D3D9, there are very few things where GL is truly easier. My transition from GL to D3D9 has been rough at best, mostly because GL actually shaped my line of thinking to its model. In the end this does not matter: D3D11 provides a much better environment for learning and it's not like writing a transform shader is a big problem.

I think the state of GL is pretty sad. Not as sad as it was a couple of years ago when Mt. Evans was not released and they gave us scraps but almost there. Well, at least they have feature parity. No wait, they don't (CS not part of the API as far as I understand).

Bingo! Are GL and CL the same thing? No. Can they interop? Sure, that's pretty different thing from CS anyway. Learn an ecosystem instead of an API. And write the code interfacing them.
BTW, shader driven D3D is column-major as well. But anyway, personal opinions are thing we could discuss a long way.

These days there's very little difference at API level between modern OpenGL and modern D3D. They work in very much the same way and you'll be writing the same kind of code with each.

D3D has, at least since version 9, been column-major internally, and the D3DX matrix library has always been able to generate either LH or RH co-ordinate systems. These are complete non-issues.

OpenGL is certainly easier to get a first triangle on the screen, put a texture on it, light it and make it spin, but that pre-supposes that you're using immediate mode and the fixed pipeline. If you want to write high-performance code with non-trivial effects things become a lot more equal, with D3D edging ahead a little owing to less legacy API cruft and a cleaner, more sensible design (no bind-to-modify, separation of buffer/offset from vertex attrib definitions, explicit registers in shaders, etc). Nothing insurmountable, but somewhat annoying to have to deal with.

D3D still has better driver quality and better tools. Documentation is a mixed bag - the D3D9 documentation is excellent, D3D10 is mostly alright, and D3D11 has it's moments but can be quite dreadful in places, with some fundamental errors that persisted for far longer than they should have (I'm specifically thinking of DstX/DstY/DstZ in CopySubresourceRegion here). The GL documentation is quite good but lack of a downloadable SDK and searchable help file hampers it, IMO. The spec PDF is something, but not enough.