I did Atkins back in the mid-90's. I overate protein (not realizing that protein isn't really unlimited) and didn't lose to where I wanted to. Switched to PP at that time, made it to my lowest weight ever. I never really felt that my cravings were under control, however, b/c I was eating problematic foods for me (grains, mostly) b/c the plan said I could have them. Still, with PP and lots of exercise, I maintained a pretty good weight for several years.
Fast forward a few years. Marriage, babies and depression brought my highest weight ever. I went back on PP and lost nothing in a month. GAve up. Saw new highest weight 1 year later.
Tried Atkins...I needed a plan taht did not allow problematic foods of any kind. Not leaving it to your choice, but saying NO WAY on the grains. I resolved to not overeat protein, remembering that protein was a problem before. I have lost back to weight where I though I was so fat all those years ago and started Atkins back then....
All that to say both plans are GREAT. PP was great for me b/c of the protein requirement (for me, it was a limitation of protein, since I tend to overeat it...) which was very helpful. Atkins is better for me b/c of the initial severe restriction on foods that are a problem for me, like grains and sugar (both of which are allowed on the old PP plan as long as you count them in your daily tally, and I was not strong enough to just eliminate them on my own like PPLP Purist. I had too much "choice" with PP/PPLP if that makes sense). That restriction has taught me how to get along without them.
I consider myself to be doing Atkins with PP/PPLP influence.