I'm not on either, I'm on a program called Lean for Life, but from what I know about South Beach, it looks similar. I think on SB you can eat more veggies and fruits. However, my program requires the body going into ketosis, similar to Atkins (I'm not sure if SB uses ketosis).
If you're trying to decide which is better for you, I think these are the main factors: with Atkins, you are likely to lose faster IF you can keep yourself away from the veggies and fruits (and of course starches) long enough. SB will probably get you losing at a pretty good pace, but probably not quite as fast as Atkins.
Atkins allows you to eat more fats like mayo, cheese, pork rinds, etc. But I don't think you're allowed fruits on Atkins for quite a while.
So I guess you have to decide that if you go with Atkins, you can lose the weight fast, but you might have to give up things like fruits and some veggies for a while, but you can eat "fattier" things, but if you never liked fruits much in the first place, then that might be for you!