Nightmare mode is just like normal mode.
The only existing difference is that all enemies are faster (like when they have been frozen once) and Igor runs faster (like with

). Also, as far as I know the fireball appears earlier.
One thing that has been discussed a while ago is the question whether Nightmare Mode makes better items appear. I have noticed that in my past Nightmare games I got far more good items than I normally would. More than one coeff. 4 item a game, in every game at least one item that is missing in my fridge (this game the Candy Cane and another Old Rusty Key), and quite a few coeff.5 items (BoS, green mushroom...). I can't believe that this is only a lucky phase - I hadn't got any interesting items before in weeks and now more than one per game...