The Steam version (Global) has toggled modes, so you can switch among them in any station. That was the big change from the original in that regard; there's no 2nd and 3rd playthrough. Rather, one can choose difficulty on the fly.
This should affect all areas you haven't been to during that game session; if you clear an area on, say, Normal, and want to run it again on Nightmare/Hell, you'll need to return to a station, reset it through the reset button on the minimap, and re-enter it.
Though I've found that button to be broken, so resetting areas should be possible by relogging.
As for drop rates, they're affected by both enemy level (in terms of what item levels drop) and by difficulty (in terms of item grades). Difficulty mostly affects the drop rates of
bosses, but
regular enemies are also affected. More Champion mobs spawn in higher difficulties, which practically increases the amount of items gained.
