Okay, had a moment to sit down and think about this properly. The quick answer is that it just seems to work better and it gives me results that I like. Obviously that's not the 'better' reason I alluded to.
The reason is that the dodge and burn tools move the shade of the area they are applied to up or down
from their current value whereas if you lay down a strip of a different colour with the paint tools it just gives a line of that colour. Obviously you can use the airbrush tool so that it only lays down a light opacity of that colour - which is close to the effect of the dodge/burn tools. To see the difference I did a little test:
dodgeBurn2.jpg
Now you can see that when I use dodge to lighten the mid-grey, it has roughly the same effect as using the white to lighten it. Equally, when I use the black to darken the mid grey, the results are comparable. However, the interesting result is where the two overlap. In the case of the airbrush, drawing over a white region with the black gives me a black line, darker than the mid-grey I started with, and much darker than the white. In the case of the dodge/burn tools, when I draw over the dodged area with the burn tool it darkens
the tone it is applied to. In this case that tone is lighter than mid grey, so the burn tool moves it back towards mid grey.
Essentially the difference is that the paint tools are absolute. When you paint a black line, it lays down black, no matter what the underlying colour is. You can make the black less opaque so that its effect is less pronounced, but what you are painting knows nothing about the colours that are already there. In contrast, the dodge/burn tools are relative. They take the value that exists already and shifts it. This depends entirely on what is already there. I find that the second method gives smoother shadows and lighting than the first, and is more forgiving of a more quick and dirty approach to laying down light and shade.
Oh, here's the effect of the layer I posted above when applied to a textured layer. You can see that the black I painted on the overlay layer (on the right hand side) is very pronounced, whereas the area which was burned and then dodged is pretty subtle.
dodgeBurn3.jpg
I hope that makes sense?