You're viewing different simulations of a custom ink. It's impossible to get an accurate representation of the actual ink so you're really seeing an approximation. On screen you're possibly seeing an RGB approximation, whilst InDesign and possibly a PDF may be showing you a CMYK conversion (which in itself is an RGB approximation of a CMYK color – since you're viewing it on an RGB screen).
In truth it doesn't matter what is on your screen or what is in your file – the printed color from a Pantone has nothing to do with your file but what ink the printer uses. You could really supply your file to the printers with any spot color and as long as you tell the printer to use the correct ink you're fine.
If the color has been specified to you then you're using the correct color, forget about what it looks like on screen (what you really need to see how it looks is a Pantone color book).
As for C vs CP, Pantone explains the different suffixes on their site:
C = coated paper, i.e., PANTONE 185 C
CP = coated paper, process simulation, i.e., PANTONE 185 CP (used in PANTONE+ COLOR BRIDGE COATED).
So they are (kind of — but not really) the same color; C is your actual spot color Pantone ink, wheras CP is a process (i.e. CMYK) simulation of that Pantone ink, so isn't a spot color and should be treated as any other CMYK color.