In general, the Exposure slider can be considered to be the software equivalent of changing the ISO on-camera -- you are changing the amplification applied to your base exposure, so you are shifting things linearly back and forth between darker and brighter.
Brightness with some software applies a bit of a curve -- it brightens darker and midtones but tries to apply less to highlights. Contrast basically makes brights brighter and darks darker to "stand out" against each other.
I adjust Exposure when it's called for, typically to the point where highlights are not blown (although a few little spots my show up as blown) and I use the highlight recovery tools to tone the highlights down to show detail and tonal variation.
If need be, I use some shadow fill. I'll use brightness if need be. I'll usually use a fair amount of contrast. Usually at some point I'll revisit those things to balance things out, including setting the "black point" to some pleasing place. Saturate to taste, sharpen, cook until well done, and out comes an image!