(sorry if there's already a thread for this: I didn't find it)
I think I know how to describe the different feeling of shooting with a tool such as a Leica M9: the camera makes you work to get your pictures, giving you just enough help so you can be successful, but not too much so that it is easy; and that makes you work harder to get your pictures, and think harder about them while you're shooting; and in the end, it makes you feel you take the pictures, not the camera
of course, I can't get myself an M9 either, but I live the "Leica experience" in a low-budget way by using vintage Leitz glass on my Canon DSLR; because my theory is that it is not the camera that creates that experience, but the (amazing) glass and the extra work and love required to use it
maybe someone with experience with both setups can chime in and confirm if the glass is the most important part, and how many extra nirvana points are actually supplied by the body
my recommendation to anybody seeking that "experience" without having to spend a small fortune: get yourself a vintage Leitz Elmarit-R 35mm f/2.8 (if you camera is APS-C) or a Leitz Summilux-R 50mm f/1.4 (if your camera is full frame); they should cost between $400 and $800 each
(yes, I like the Elmarit-R 35mm f2.8 better than the faster but softer Summilux-R 35mm f/1.4)
that 35mm gets a lot of praise whenever I post any videos shot with it, usually while talking about different aspects of the image; last time, talking about the color-correction tools, then, out of the blue, someone posts: "I’m loving that 35mm lens. Looks so pretty."
see it in action, for example, here (for the bits in the mountains):
http://vimeo.com/28918901
edit: keep in mind, in any case, that vintage glass is always a lottery, you may get a lemon; I've got 2 bad lenses, plus 2 that could have been better, out of 11 vintage lenses bought on ebay and on a specialty online shop; I think that means I've had pretty bad luck. but in exchange for that I've gotten quite good at repairing vintage lenses

see what a lemon means here:
http://www.similaar.com/foto/lenstests/lenstestsc.html