Shorter Flange Focal Distance (FFD) (the actual distance from the sensor to the lens) means that wide-angle lenses can be made smaller, cheaper, lighter, and sharper than what is physically possible for DSLR's. This applies to other lenses as well, to a lesser extent.
Fewer Moving Parts. In any digital system, the mechanical (moving) parts are always the most prone to failure. Because they have fewer moving parts, mirrorless cameras should last longer on average (assuming the same build-quality for everything else).
Audio-free. DSLR's are pretty noisy (sound-wise) when shooting, due to the mirror moving up and down. The noise can sometimes ruin the shot (eg. scare away the animal you're shooting), or get you in trouble (eg. when capturing the silent-moment for someone who has died).
Cheaper. All other things being equal, a mirrorless camera should be cheaper to produce, simply because it has less precision-parts. Lenses of similar quality should also be cheaper, because they can be made with fewer/simpler lens-elements due to the shorter FFD.
What this means is that, if you have $XYZ amount of cash allotted to spend on a camera or lens, having a mirrorless will allow you to get higher-quality parts for your money (Note: This is only mildly true right now, due to supply-and-demand, but as the technology becomes more popular, prices will lower).
This is important for anyone who doesn't swim in a pool of money every morning.
Can't be used in low-light. Because there is no optical viewfinder, if the light is too low, the camera basically can't be used. This is the major issue with mirrorless cameras.
Display will never be as "real-time" as a viewfinder. However, displays have gotten so blazingly fast that there is zero practical lag (1ms or less) on the higher-quality screens. The only time I can imagine needing true real-time is when following a high-speed object, like in some sports or nature photography.
Battery life. Mirrorless cameras usually have shorter battery life, due to having to keep the LCD and CCD on all the time. As LCD's, CCD's, and batteries become more efficient, this becomes less of a problem.
Now, opinion time: Is DSLR a dying breed? I personally don't think so.
Right now, mirrorless cameras have a perception of being non-professional, so Pros don't want them. However, without Pros willing to buy them, the camera manufacturers are hesitant to produce super-high-quality mirrorless cameras, because no one will buy them. It's a sort of bootstrap problem; but as the technology slowly gains acceptance, the issue will solve itself.
Even once the technology catches on, though, mirrorless cameras will supplement, rather than replace, DSLR's, since each have their own advantages and disadvantages (above). So, what you will probably see 10-20 years from now is Pros with both mirrorless and DSLR cameras in their repertoire, for different occasions.
