Hi, I've done both.
With Atkins, I lost a stone in the first month, but then it was quite slow and I didn't stick to it well because of that. In the end I was just staying the same. This wasn't as bad as it sounds because before that my weight was just increading all the time no matter what. OVer a year I came down from around 225 to 190 lbs.
I was pretty desparate when I found CD and had been thinking of surgery - I had all the brochures and been to the docs. I was sceptical and thought the weight would go back on as fast as I lost it but I found loads of research saying that was a myth.
CD was tough but I loved the fast losses 7lbs in the first week and then a stone a month no matter. It worked. It was like magic (I'd put on weight with Slimming World!) and I couldn't believe it that if you stuck to it you really lost the weight.
Also, it was good to have a break from my relationship from food. It allowed me to take stock of my life and binge eating issues without ending up in a packet of crisps!
I started 2 years ago and I'm 10lbs over my target weight at the moment but am dealing with it.
When I came off CD I went onto low carb to mantain and it works well.
hope this helps you with your decision and best of luck with the weight loss. This site and the support I got and still get was a big factor in my weight loss.