RAW gives you access to the image the camera took. With the right tools, you can then manipulate that image: correcting chromatic abberation, correcting lens distortion (e.g. via DxO), sharpening, and even changing the apparent exposure. All changes are lossless and reversible.
With JPEG, first you're dealing with something the camera created, using the camera's algorithms. For example, the camera's idea of how much sharpening should be applied, the camera's built-in dynamic range shaping, color gamut and so on. Some cameras do a good job, others get it hopelessly wrong: e.g. the Ricoh GX200 takes great pics but makes terrible jpegs. Finally, the image from the camera has already been lossily compressed, and each time you manipulate the image a little more quality will be lost. Only a few tools can even rotate a jpeg without more loss. Even Q=100 gives about 2.5:1 lossy compression. Interestingly, near Q=100 as compression artifacts increase, saving a file multiple times reduces its quality yet increases its size dramatically.
So if you ever think your tools can do a better job of making a jpeg than the camera's built-in algorithms, you're usually right. That's why RAW (or DNG) is important: access to what the sensor saw, rather than what the camera thinks you want to see.
Most cameras, of course, can shoot both at the same time, so you get a jpeg to look at and send people, and a RAW file to archive (on the assumption you're not going to edit every photo you ever shoot).