I've always struggled with this and more and more often it causes me issues in the stage when I've received payment and then they client tries to print their design. Especially with my designs that are full of color the files I email them look great. And when they are printed they look like they have been bombed. And this seems to happen randomly for no reason. Even if I create two logos exactly the same way but perhaps with different color schemes: one will print exactly the same as it displayed on the computer and the other will be fried?! Please help me out guys...I feel like a complete idiot and am losing my mind!
On-screen representation is NOT an accurate representation of the final printed color. Even custom calibrating your monitor to get accurate color rendering will not always guarantee a match with the actual printed color.
There is such a large difference between the way the color models are rendered, this makes using a monitor for color proofing difficult. See this answer: https://graphicdesign.stackexchange.com/a/38421/21203
A pantone book / swatch will offer you a much more accurate color representation. Even then, have the printer provide you with a sample of the pantone color printed on the substrate you are targeting to ensure the color prints accurately. Substrate or ink variations, pantone book / swatch degradation, print run inconsistencies and of course, lighting environment changes, will all cause shifts in color balance and accuracy. A hard proof will let you compare the final, printed color against the original pantone color you chose.