I feel bad leaving anything completely unsaid, and I regret not being able to make a full statement on the subject earlier. Below I've provided a few links and some extra information for those genuinely interested in the topic. I've tried my absolute best not to over- or under-present any aspect of either DirectX or OpenGL.
http://www.gamedev.net/reference/articles/article1775.asp
-Direct3D vs. OpenGL: Which API to Use When, Where, and Why
This is an old article (2002), but it is a good introduction to the differences between the two and the history and conflict between the two. The state of things now (as in right now) is basically equivalent to the state of things in this article. When DirectX 9 came out, it was more feature-complete and had better support for various things than OpenGL. However, OpenGL has very modular updates and simply added the missing features and support in after the fact.
As it is now, OpenGL 2.1 (the current version) is equivalent in feature-completeness and support to DirectX 10 (this is not me stating this, I haven't done API programming in years; this is direct from a guy that has used both OpenGL and DirectX since DirectX 5).
For those of you that are interested, here are some slides from nVidia's Michael Gold about the upcoming OpenGL 3.0. A lot of things have been streamlined in this major update.
http://www.khronos.org/developers/library/2007_08_siggraph/siggraph%202007%20bof-gl3%20overview.pdf
(Now's a good time to uninstall Adobe Reader and switch to Foxit Reader, by the way

.)
http://en.wikipedia.org/wiki/Comparison_of_OpenGL_and_Direct3D
-Another Comparison
Here's a fairly good, slightly more modern comparison with (most of) the fanboyism (of both sides) left out.
Keep in mind that the history of Microsoft, DirectX, OpenGL, nVidia, ATI, and lots of others are all very intertwined in a complex web. They all, in some way or another, depended on one another and fought for and against each other in complex moves.
------
Concerning my earlier comments, I do not think DirectX is a bad platform. I simply am more philosophically aligned with OpenGL and feel it is a better long-term option for everyone involved (apart from, perhaps, Microsoft). DirectX, despite being a child of Microsoft, is actually well-developed and has become better over the years despite early setbacks. It is a good platform, and Microsoft has streamlined its use with a series of tools and cross-compatibilities with other things like .NET etc. OpenGL doesn't have the benefits having lots of streamlined tools (for games) and lots of popular engines with good support (though John Carmack is doing his damndest to stop this; check out idtech5).
Both platforms, in a vacuum, have no large differences even (amazing what hardware has done to drive them along the same path). As it is now, though, DirectX is as much a marketing brand as it is a development platform. One curious point today is that DirectX is basically equivalent to Direct3D. DirectSound and DirectInput and all that is fairly deprecated today and only commonly used with things like rootkit implementations (Starforce etc.). DirectX supporters used to be able to say that it was a whole platform while OpenGL was only a graphics solution, but these days it just doesn't fly, and DirectX has to compete on its graphics alone basically.
The reason I am personally more aligned with OpenGL is that it allows for more overall freedom for developers and for users in the long run. If DirectX dominates the market somehow in a few years, people will have less and less choice as developers are pushed more and more to make DirectX games as opposed to OpenGL games. Everything but Windows gaming will suffer (and that might even suffer if Microsoft gets comfortable enough [see Vista/DirectX 10]). I don't like the idea of being forced, some time in the future, to buy a new operating system at the cost of hundreds of dollars just to be able to play some games. As Microsoft showed with Halo 3, they are willing to go this route. This is just my personal view of the matter, and others are welcome to disagree for their own reasons. But saying things like 'DirectX > OpenGL' demonstrates a lack of understanding and only leads us to heated debates that will eventually get nowhere.
I encourage developers to use OpenGL because of this. If more developers show interest in OpenGL, we will end up with better methods, better examples, better tools, and a better OpenGL community. This is better for everyone in the long run (in my view), because it leaves us with more choices, more ways to accomodate more people.
-----
As Ceylin said, the two are just graphics API's. One isn't FUNDAMENTALLY different from the other.
The specific example he cited is true, and it just demonstrates that Microsoft has made a poor decision in gimping DirectX 10 to be 'Vista only' when it doesn't really have to be that way. That's not a mark against DirectX 10, but it just demonstrates (in my opinion) that developers can't depend on Microsoft to do sane things forever, even with something as excellently-developed as DirectX.
As it is now, OpenGL doesn't lack anything compared to DirectX except EXAMPLES of real IMPLEMENTATIONS. No one's made a 'Crysis' (or pick another game for a better example, if you like) yet for OpenGL (though John Carmack is working on this). While this doesn't really matter intellectually (such a game can be made with OpenGL), it matters for the visibility effect since OpenGL is not something that is marketed.
-----
To sum up, both are good platforms for developing games with DirectX having the ease-of-use advantage (because of Microsoft tools and 'integration') and OpenGL having the cross-platform and (in my view) future-proof advantage. In a vacuum, the choice between one and the other is basically a coin toss. But when considering the long view, I'd rather that games run on an open standard instead of at the whim of Microsoft. Others may disagree.
Hopefully this post has been helpful and has positively contributed to this thread. Yes, it is 100% on-topic. Gaming on Linux basically translates to implementing games in OpenGL.