The mirror means that you have an optical viewfinder (rather than an electronic viewfinder, which is a small LCD screen). I find this much easier on my eyes to look through. However, electronic viewfinders have their benefits. What you see is what you get. The photo comes out exactly as it appears in the viewfinder. Also, with the press of a button you have instant magnification. It's easier to see if the focus is correct (which is very useful if using older manual focus lenses). But virtually everyone would prefer a good optical viewfinder to an electronic viewfinder if you are using the camera a lot.
A typical SLR also focuses a lot faster than a mirrorless camera. But mirrorless camera are said to focus more accurately. Mirrorless cameras are also improving on their speed.
Getting rid of the mirror has advantages. The main one being that it decreases the size and weight of the camera. There are some other thoertical advantages. Without a mirror, there is less vibration when taking a photo - possibly leading to sharper photos (but being lighter, they are a bit harder to hold still). You should be able to design great wide angle lenses (but apart from Leica - and do they count? - nobody has really done this). With less moving parts, you should also be able to build a tougher camera (but once again, nobody has actually done this). The Nikon V1 also shoots at 60fps. I don't think that Canon SLRs currently do this (but in theory they could add this capability by operating in mirror up mode).
Oh, and then there's batteries. I think I'm currently getting over 1500 shots per charge with my 1Ds MkII. I'm lucky to get 300 with my EPL-1. To keep the camera size down, the batteries are tiny. Its easy to carry a spare, but just another thing to consider.
For what its worth, over christmas I took a few weeks off and hit some typical tourist locations. As always, nearly everyone had a camera of some sort. But this was the first time I'd noticed a lot of the Pens, Panasonics, and new Nikons. Its not reflected in the sales figures, but it would seem that the people that actually get out there and use their cameras tend to like the new breed of mirrorless cameras.
Therefore, if you are looking for a light, portable camera for general photography and aren't interested in a lot of lenses, I think its a dead heat. In fact, assuming similar sensors, image quality would be the same. If a non-camera person asked me for a recommendation, I'd give the nod to the mirrorless camera.
If you are after something with fast focusing, that you will be using all day, with a lot of options for lenses, then the traditional SLR is the clear winner. At least for the next two years.