In Lightroom the Exposure adjustment operates mostly towards the right hand side of the histogram, leaving the lower mid tones and shadows relatively unchanged. The Brightness slider operates very much on the mid tones, brightening the overall impression of the image without pushing the highlights into clipping. If you hover the mouse of any of the first four adjustments (not Brightness or Contrast) you will see a section of the histogram become faintly highlighted, so you can see whereabouts the changes will have most effect. You can also select within the histogram itself and slide left and right there in order to invoke the Exposure/Recovery/Fill/Blacks sliders.
If you have a gap over on the right of the histogram and you want a lighter image then you might start your adjustment with the Exposure slider first. If you are already near or at clipping with your highlights but still feel you want to lighten the image then try the Brightness slider rather than Exposure.
If you have a flat, drab image, such as one taken on a foggy day, with a strongly centred histogram and gaps on both sides, then you might want to pull up the exposure slider to brighten the top end and also pull up the black slider to darken the black end. You can then use the brightness slider to position to mid tones wherever you need then and then tweak contrast to best effect.
In DPP you only have the one slider in the raw tab, called "Brightness", I believe it shifts the entire histogram left/right as though you had actually shot with a different exposure within the camera.
I don't know about other software.