With film you are worried about zero density: under exposure (shadows) with print film and overexposure (highlights) with transparencies. Technically you could burn through too much density, and still recover details, although sometimes this is easier said in theory than actually accomplished in the real world.
With digital you have to worry about both ends of the tonal scale during exposure. One step beyond either edge of the histogram is the same as zero density or 100% solid density.
With digital I shoot raw, and generally follow the expose to the right school of thought. I am amazed at the detail I've been able to pull out of dark shadows, but I still think it ends up better quality, to darken raw files rather than lighten them. For instance it's my experience that I will end up with cleaner (less noise) looking photos shooting ISO 1600 and getting the exposure right on or slightly overexposing it, than I will slightly underexposing ISO 800.