During a diet based detox you are still eating fairly normally, but you are eliminating many of the unhealthy substances and replacing them with healthier ones. This way you reduce the load on your digestive system and give your body more time to detox. You are also slowly replacing bad quality nutrients with higher quality ones.
I have personally never used diet based approaches for detoxing. For me they are more like maintenance between fasts. I know that there are some good programs out there that will give you good results.
If your goal is only to eliminate acne then many of the diet based approaches should also work for you. My only concern with them is that they don't activate the same natural healing process in your body that fasting activates. So I really don't know how deeply they will cleanse your body.
Good thing about diet based detoxes is that after you are done with them it's easier for you to continue eating healthier -since you have already gone through the change process.
Another thing I'm worried is that how many people can actually follow through such a diet. With all kinds of cravings urging you to eat what you used to eat it may be difficult to go through them. That's why I feel that it's easier to bear few days of discomfort at the beginning of a fast and be done with it.