Alot of things.
You can perform non-destructable fixes on different things.
Like exposure, WB and sharpness.
Ofcourse you can do that with a JPEG too, but the RAW is just pure info, so you can always change, change and change on your different measurements.
I must say that my explanation was a little bit fishy, and it doesn't cover all the features of RAW shooting. But I will suggest THIS LINK if you want to dig deeper.
When it comes to RAWshooting myself, I do it all the time. Ofcourse, I evaluate the Situation, do I really have to shoot in raw? Do I need those features. In my artwork, yeah I need Raw, but in like music-photography I always use jpeg. It's quantity vs quality.
I first tried raw year ago with my Olympus C5050z. I took pictures in JPEG and in raw and compared them. I saw no difference.
Now I shoot only raw. What changed? The first thing that changed was that I got a DSLR and started taking photos in situations that did not have optimal lighting. I didn't do that with my old camera.
The second thing that changed was raw programs improved immensely.
The non-destructive nature is a valid point. When I first started I tinkered with JPEGs and basically ruined them because they were edited and stored so many times as I was learning how to edit. Although I'm still learning it's no longer a problem.
My opinion is that what is best for you is determined by what you like to shoot.
- raw is not pre-sharpened, so you can decide about the level of sharpening a scene needs (ok, you could also switch sharpening off for jpg)
- Whitebalance is just one click and you do not have to care about it while shooting
- RAW gives you the bit depth per colour which is provided by your sensor. hence later editing will not lead to posterisation effects so easily (no steps in what is supposed to be smooth colour gradients)
- afore mentioned point allows for more brightening up of shadows or enhancing contrast in scenes which were not easy light-wise.
- with RAW you decide how to "develop" the data, not the "lab" in your camera with its rigid pre-sets.