The most significant problem of the traditional method of difficulty scaling is that it is completely opaque to the player until he or she actually chooses a difficulty setting and plays with it for a while. You can't really quantify the differences easily, and can't describe the differences to the player without revealing a lot of information about the inner workings of the game system.
However, there alternative means of varying difficulty than just manipulating damage and enemies. In my last blog post, I described how the inclusion of permanent loss in the Fire Emblem and Megaman series affected game difficulty. Since it has a noticeable effect on game difficulty, it means that it can be manipulated in order to adjust game difficulty and perhaps form the basis for various difficulty settings. For example, the possibility of permanent character death is what makes the Fire Emblem series so difficulty. It is not inconceivable to imagine adding a new "Easy" difficulty setting to a Fire Emblem game where characters simply don't die permanently or have a finite number of extra lives. This would certainly make the game somewhat easier, by allowing the player more leeway to make mistakes. And this is just one example of how it is possible to adjust game difficulty by modifying various different gameplay systems.
In fact, two recent Fire Emblem games did use nontraditional means of adjusting difficulty. In FE: Path of Radiance, the Easy and Normal difficulty settings did not have Fog of War on any maps, while hard mode did have it on specific maps. In FE: Radiant Dawn. Easy and Normal modes allow the player to make a mid-battle save, while the player can only suspend their game on Hard mode. Another example is from Megaman Zero. In that game, a basic feature of the game is that the player can build up weapons by using them and unlock new techniques, such as the charge slash. However, on Hard mode, the player cannot build up weapons, and is stuck using basic techniques. Coupled with bosses who have moves they do not possess in Normal mode, and the game becomes much more difficult.
The advantage of using techniques like this is that they are obvious to the player. These differences can be explicitly spelled out to the player in the manual and can have an obvious effect on difficulty. They also allow the developer to allow the player to control how "hardcore" a game is in ways that cannot be done with traditional scaling of enemy power.