Thanks for the excellent question WingZero30.
The short answer to that question is that they're on an entirely different level. Having seen demonstrations of Samsung's OLED monitor technologies in Asia last year I can honestly say that it is an instant "wow - this is different" kind of experience. The overall image is just so much more natural looking and beautifully crisp and vivid.
Because they do away with the backlight and have nice coloured organic light emitted directly from the pixels themselves you have much greater control over the colour and luminance of each pixel. What this means is that instead of having a set brightness across the screen (or in areas as you get with some more expensive "local dimming" LED backlights) you have a set brightness for each pixel - some could be off to give the purest black, whilst adjacent pixels could be on to give the purest white and of course any colour in between. You can think of it as a kind of highly efficient, much higher resolution and flicker-free alternative to plasma.
Because there is no backlight, the colour gamut is dictated by the pixels themselves and is therefore exceptionally broad. It can also be much more tightly controlled than in existing technologies, yielding superior out-of-the-box colour accuracy. You also don't have to wait for any filters to operate when producing colour so the response times are almost instantaneous (in the order of <0.01ms). The direct emission of light from the pixels, with only a few thin layers of material infront, means that the image has fine colour and contrast at practically 180 degrees.
The efficiency of OLED technology is being improved all the time, but some speculate that once they are widely available you could have a 24" OLED monitor use around 1W of electricity for every 100cd/m2 of brightness (so 4W for a sustained 400cd/m2 brightness). This is oversimplified really, but efficiency is set to be a real strength of OLED technology.
Aside from efficiency; the manufacturing process itself is constantly being refined. New spraying processes, modified organic materials, thinner cathodes and TFT backplane materials can all improve the characteristics (and impotantly, lower the cost) of OLED monitors further. Hopefully in a few years time they will be ready for the mainstream monitor market and they will find their place for a multitude of applications - both professional and home uses.