These processes were meant to APPROXIMATE the selected color of how it will look when printed. Unless you have gone through an extensive color calibration process these devices each shows colors differently. You can take the same image on 5 monitors and it will look different on all of them. The same will occur when printing out to 5 different laser/inkjet printers. Each device has different degrees of calibration that are possible.
A monitor displays color in RGB (red, green blue). Colors are created by adding red, green and blue light (additive process). Offset printing will typically be either CMYK (cyan, magenta, yellow and black) or PMS Colors. Inked paper absorbs or reflects specific wavelengths (subtractive process). Cyan, magenta and yellow pigments serve as filters, subtracting varying degrees of red, green and blue from white light to produce a selective range (gamut) of spectral colors. PMS Colors are specially formulated colors that don’t always have CMYK or RGB equivalents.
Both RGB and CMYK can only represent at best a relatively small part of the total color space, and there are some colors that can be produced in one but not in the other and vice-versa. The range of color that a color space can produce is called the gamut.

Another element that affects the way the color will look when printed is the stock. An uncoated stock will absorb more ink creating a heavier image. On a coated stock the ink will typically sit more on the top of the surface. The coating on the stock had a different reflective property creating more vibrant images. If the stock has a color to it the inks will be affected by the color. If you print a light blue on a cream stock that light blue will have a tinge of green to it. When using special or colored stocks it is best to ask for ink drawdowns on the actual paper. This will give you a better idea of how the actual color will look. Something to remember with drawdowns – an ink draw down is created by putting the ink onto a paper by hand without water — when you actually get on press and add all the other elements in there can be a slight variations when on press.
For CMYK offset printing we recommend a matchprint proof to show how the color will print on press. The matchprint device has been calibrated to match the printers press. Each press will print a little differently (some may be heavier reds, some may be heavier cyan.) Matchprints can be calibrated to show the difference in what happens to an image on Coated vs Uncoated stock. This will help to eliminate any surprises on press.