Being a fairly avid D2 player, I can tell you that it works in roughly the same way. Just like in D3, as you go up in difficulties, creatures become much stronger. In D2, many of them gain immunity to specific elements, and the likelihood that you would encounter a unique (yellow/gold monsters) or pack of champions (blue monsters) is increased. Additionally, uniques and champions have more modifiers when you encounter them (similar to in D3).
In much the same way, the Acts also scale in difficulty. In fact, Blizzard did a very good job of making Act 5 harder than Act 4 of the same difficulty, but not so much that it was much harder than Act 1 of the next. A lot of this comes from the fact that as you increase in difficulty levels, a lot of your stats are reduced. For example, in Nightmare, all players suffer a -50% reduction to all resistances. Additionally, life steal effects are cut to a quarter. When you go to Hell, the player starts with -100% resistance (yes, it was completely possible to have all negative resistances, which would, as you'd expect, amplify damage taken from that source), and life steal effects were cut to 1/8.
Another big spike in the difficulty was also mentioned above: the more common appearance of uniques and champions. In Normal, it is unlikely that you will find more than 1 or 2 packs of uniques or champions in any given area (for example, the stony field), excluding super uniques which always appear in the area (like Rakinishu, Blood Raven, Bishibosh, etc). However, in Nightmare, it is not unlikely that you will simply encounter 2 packs of uniques/champions at once. Then you might have to deal with something like "Cold enchanted, extra fast" at the same time as dealing with "Lightning enchanted, multiple shot". That, along with natural monster immunities (for example, the entire Fallen tribe becomes completely immune to fire damage in Hell mode) makes Hell mode much more challenging than even Act 5 of Nightmare.