One of the major features of new DX versions is to do the same things that we've been used to, but a lot faster.
That can result in two things.
Much better performance, meaning hardware with less raw power can run games that look nice, just as well as the previous generation (for example), or they can introduce more features that are more efficient and keep performance to an acceptable standard.
I've been saying this for 5 years, DX versions offer NOTHING, at all, that can't otherwise be done. However they can offer acceleration of certain effects, or less over head and as Kyle listed, you can use this to either give more performance, or with the saved performance, add more effects.
The look of a game is 100% down to the programmers design capability, nothing more, nothing less. You can make Metro2033 look the same in dx11 and dx9, however its likely with higher res models, and more advanced shaders that dx11 would run faster and offer higher quality shadows, lighting effects and so on.
A texture is a texture is a texture though, you design a billboard, its a little drawing, its saved and then drawn by the card when you walk past the billboard in a game, no matter what type of lighting is used to light it, no matter what kind of shadow it casts, no matter the card it looks on, the underlying look of the billboard, is identical.
Games look the same under dx9/10/11 excluding a couple of VERY minor options, because 98% of the work done in a game is NOT DX specific.
Regardless of DX version the 58xx series is simply faster than any specific "DX10" card ever made, because it has more shaders, more rops more TMU's than any DX10 card ever made.
The single most important function of DX versions being changed, is a general push forward of idea's, support and functionality that allows developers to have a pretty specific list of things to support, do and have ready for games to be made in the future. IT also, contrary to what people say, gives a fairly stable set of rules/programming idea's with which to code a game that will remain compatible with almost any system.
people go on and on about consoles being easy to program for because the hardware doesn't change. Hardware is almost irrelevant, programmers program to SOFTWARE rules and DX versions and the DX API give a very specific set of rules to work for, if you remain in the rules it shouldn't have anything but a few issues working on a DX card.
So it gives us an important place for programmers to aim for and newer versions give programmers a direction to aim for in the future.
In terms of end user, the DX version your game uses, means NOTHING.
