When going through my workflow I find that at times I go to "exposure" to lighten or darken certain images and at other times I use "white." There doesn't seem to be a logical pattern to follow. I would be interested to hear how others approach this question, i.e., when do you use one, when the other, and when both?
Adjusting exposure makes the adjustment to the entire image. When you are using the Whites adjustment you are making adjustments to only the whitest/brightest parts of your image. Also, using the White slider in conjunction with the alt/opt key enables you to set the appropriate white point for your image. Conversely using the Black slider this way can help you set the black point properly.