Why do the RGB values specified for a Pantone color in the Formula Guide (Color Bridge Coated) differ from the RGB values specified for the same Pantone color in the Pantone color manager software?
It is because the RGB in the Formula Guide is based on the C (the spot PANTONE color) and the one in the software is based on the CP (which is CMYK).
You can test this by making a photoshop document (RGB) and fill two areas with C and CP color of the same number. I haven't tried it with 3514, but in most cases they will produce different colors and different values.
Edit: CP is a simulation of the C (the spot Pantone color) in a process separation – which most of the time, as in this instance, is CMYK. That is why CMYK values of CP is the same as the simulated C. If you were to print 3514 in CMYK, you could pick the CMYK value or the CP color and get the same result. The RGB values are different because they are based on different colors – spot and process.
According to Pantone's website (Pantone > Help Center > Graphics Matching PANTONE Color Manager data to PANTONE COLOR BRIDGE guide), if you export swatches from the spot Fan Deck as RGB (not LAB), you should get the same RGB/HTML values as those printed on the Color Bridge guide. I haven't tried the export, but if you double-click on individual swatches in Color Manager, they are NOT the values from the printed guides. Changing from no color profile to sRGB does not help.
So, I still manually create my Web swatches using the Color Bridge information, as there doesn't seem to be a solution from Pantone. Not really surprising that a printed ink company doesn't make screen representations a priority.