Pantone vs CMYK — two different physical systems
Pantone (specifically Pantone Solid Coated, the most common brand-identity reference) is a system of pre-mixed inks. Each PMS color is a specific blend of base pigments — 14 base inks in current Pantone formulations. When you specify PMS 7637, the printer's ink lab mixes that exact formula and runs it as a single solid color on press. The color is exactly what the swatch book shows.
CMYK (cyan, magenta, yellow, black) is a 4-color process. The printer doesn't mix custom inks per job; instead, the press lays down halftone dots of the four base CMYK inks in patterns that the eye blends into the target color from a normal viewing distance. CMYK is cheaper, faster, and works on every press. But it can only reproduce colors that lie inside the CMYK color gamut.
Gamut is the range of colors reproducible by a given color system. The CMYK gamut is smaller than the Pantone gamut, which is smaller than the human-visible spectrum. Pantone has rich oranges, deep greens, bright magentas that CMYK cannot reproduce in-gamut.
About 50% of Pantone Solid Coated colors cannot be reproduced exactly in 4-color CMYK. Forest greens, bright oranges, rich purples, deep navy blues are the worst offenders.
