There are two different things going on here, and both are unrelated.
The first is that inks look different when printed on coated versus uncoated stock.
The formulas are the same. The only real* difference is the paper coating. On uncoated stock, inks tend to absorb into the paper, on coated stock however inks sit on top of the coating, and dry by oxidization. This results in a slighly more vivid colour on coated stock.
Secondly, for the differences between solid and CMYK colours, it's because CMYK colour is only an approximation of a solid colour. Some solid (or spot) colours can't really be achieved accurately using CMYK. CMYK printing is a different printing process from printing Solid Pantone colours. Because of this, they will never look the same. There's more info here on another similar question I answered.
I think the reason the CMYK values differ for the Bridge CMYK colours (for coated versus uncoated) is because the guys at Pantone have tried to compensate for the problems of simulating Solid colours using CMYK process printing on coated versus uncoated stock. Because it's a different process, if they left the CMYK values the same, the colours would look even more different on different stock.
*Note: I say only "real" difference, but sometimes printers alter the formulation of an ink depending on the stock or different processes involved in printing. This is not something that affects colours or that clients/designers generally need to be aware of. It can involve the use of additives or using different bases such as oil or rubber based inks, etc.