What makes you think your monitor is displaying the "exact" color you think it is? For one thing, they need to be calibrated http://spyder.datacolor.com/
Besides that monitors have a certain colorspace with better monitors able to show more color gamut. Viewing angle effects the color (much more variation on TN panels than VA or IPS panels).
If you show the same picture on 2 different uncalibrated monitors, you'll get very different results.
As others have said background color makes a huge difference in perception. So do viewing conditions. When calibrating a monitor it factors in ambient light. When printing pictures for exhibition, my print software compensates for the ambient light the picture is intended to be viewed in.
How do you intend to generate the colors on the LED, run it at full power and PWM it? That's relying on the performance of the viewers eye to render the color and you're really displaying nothing remotely like the color on the screen even if you manage to get it to look similar.
Even calibrating something as simple as your white balance, getting pure white and pure grey is very hard to do.