Bridge guides provide "process color simulations of all solid PANTONE Color". In other words, the bridge guides are all printed in CMYK to match Pantone spot colors as close as possible. You see the CMYK color on the guide along side the actual Pantone spot color. This lets you compare how close the CMYK color is to the spot ink.
Formula Guides are printed in the actual Pantone spot color. So you see the Pantone color on the guide. There's no representation of other color models. It's straight pantone spot colors. It provides numbers for other builds, but it doesn't show them to you.
That's the difference.
If you customarily use Pantone colors as just a general color picking form, but print everything in CMYK, or if you are doing a great deal of web work, the Bridge guides may serve you better.
If you are specifying actual spot colors to use on press, then you want the Formula guides so you can be sure you see the color accurately as it will come off of a press.
Really, if you just want one guide... get the Bridge. It's pretty much the best of both worlds.