Joch's note #1
For Hard/DMD bosses, not very much changes. In Hard mode, bosses have the same health as Normal ones, but deal 2.5x more damage. In DMD mode, bosses have 2.5x more health than Normal/Hard bosses, and deal 2.5x more damage. Only in a couple of cases do attack patterns change; otherwise these strategies can be used across all modes - you'll just have to endure for longer in DMD mode!  

Joch's note #2
I've assumed the player is on Normal mode or above. I strongly recommending avoiding Easy Automatic mode, which is very much lamer than DMC3's Easy (non-Automatic) mode. The game's wrestling of your controls isn't cool, plus everything has been dumbed down to a degree which butchers the DMC gameplay. Not only that, but at the end of every EA game, the game dumps you into another EA game - you can't progress onto Normal mode from EA mode. To access the higher modes, you must play Normal mode.