Look for the Color settings (Command+Shift+K) might solve your problem.
Your provided artwork and your computer might be using different RGB profiles.But few things you need to know.
RGB and CMYK values are merely an estimation of how the color would be displaye or printed. Depending on the Ink providers, different combination of C, M, Y, K produces visibly different colors.
Pantone colors are supposed to be used where spot colors are required to print (One specific color for a brand color e.g. Coca Cola red means a specific pantone color which is supposed to print the same.
A bit off-topic but even with pantone colors, if the quantity of ink becomes low, the color doesn't print the same.
With your question, I'm assuming that You are provided an artwork where some Logo and brand graphics are used with Pantone color. When you click and try to convert their swatch to CMYK, it doesn't appear the same on the screen and doesn't match on the color book either .
The smallest answer : (it won't). If you'd try to match it on screen, it might print very differently than it's pantone.
If you are going to make sure 100% of printing the brand colors with pantone, you'd have to go through separate print jobs for the same artwork, one with Pantone and other with CMYK plates. Based on your artwork, your printer might suggest which job should print first.
Depending on the nature of the print job, If it's some marketing campaign and you have a margin of being flexible, you can easily depend on the converted CMYK values, (though they're not looking the same on the monitor).
One more thing we often do is to consult the printer (Person who prints the offset printing), make the hit and trial for different CMYK values and use the ones which our printer suggests to use.