CMYK is made up of four colours: ‘C’ cyan, ‘M’ magenta, ‘Y’ yellow and ‘K’ black. When printing four color process, each color is put on the paper separately, and then layered. If you zoom in on a printed image you will notice half-toning or little dots of colour  layered over one another.

As a result of the above, CMYK colours can vary depending on the angle of application for the above colours, the paper stock, printer and several other elements. Simply, they are not a universal standard.

In order to provide a standard, the Pantone system has been created. As others explained, it is often referred to as a spot colour. It is an ink that is premixed to the colour required and printed from a dedicated plate, rather than being simulated by overprinting dots of ink like CMYK. This means that you can print a wider range of colours not achievable through CMYK, neons and metallics are two examples of this and both feature in the Pantone system. Because it is a premixed ink, it is a solid colour, it will be of a higher quality on the final print item and also allow you to ensure you know exactly what that colour is (thanks to the PMS books and examples on coated and uncoated stock).

A lot of people use Pantone inks when they either want to:

- Print a limited range of colours e.g. Just the one Pantone, in this case it could be cheaper than creating the same colour via a 4 colour method (CMYK)

- You require a colour that sits outside what's available through a CMYK process e.g. A neon

- You require an accurate representation of a specific colour at high quality, Pantone colours will allow this so long as you use a printed sample book as reference. On screen colours can still vary due to calibration so always refer to an up to date book. 

Some people believe that CMYK breakdowns are now a suitable method for overriding Pantones and in certain situations this may be correct where for reasons like the budget, such a large emphasis isn't placed on a colour being 100% accurate. It's a second best case scenario which is why a lot of brand guidelines contain a CMYK breakdown of the Pantone references. 

In my opinion, Pantone references allow a company/brand to ensure consistency across all their printed Comms and where possible the Pantone colour should always be used. It is the designers job to ensure a suitable CMYK breakdown can be supplied for instances when this is not possible.