Slimdreamer -
I've been pondering the same thing.
In essence the fat in atkins and the limited carbs in dukan work the same way - they provide enough energy to stop the body going into starvation mode, but not quite enough, so stored fat is also converted into energy leading to weight loss.
Atkins made some very basic errors - for example he thought it was the fat in the diet that supressed hunger, and research in the 2000s proved it was the protein, and so made the very high fat intake of the original version of the diet unnecessary.
Personally I think the dukan approach is far more palatable and sustainable long term, includes a far greater range of nutrients, and has a better designed transition back to a stabilised longer term diet at the end.
Plus most of us don't get the long term stinky breath problem!