Shooting RAW gives you ALL of the data that the camera sensor picks up. In post-processing you have far more options in terms of color correction, white balance correction, resolution when you crop, etc...
JPEG's are a compressed format. The camera applies some basic sharpening, contrast, saturation, etc... and then THROWS AWAY any data that the human eye can't see.
So if you have some shadows that are black and show little to no detail, chances are in JPEG, there is no way to recover that detail. In RAW, you're able to.
The problem with RAW is that it ABSOLUTELY requires post-processing on every pic. In RAW the image will appear flat tonally, it will appear soft and in need of sharpening, the colors will seem drab, etc...
If you're serious about photography, though, RAW is generally the way to go. If you're doing event photography and your white balance is off and the bride's dress has a yellow hue to it... you wanna be able to correct that. The more data you have to work with in post, the better off you are.