One of the major differences as I recall, the SBD does not restrict you to a certain carb count initially as Adkins does (at least when I tried it a few years ago I believe the first week or two you're not to exceed 20 grams carbs). SBD was much more flexible and easy to live with for me.
You will notice in the list of foods not to eat the first two weeks, sugar is the thing to stay away from, either refined or in fruit. Like Adkins, this is to rid your system of the stuff and eliminate cravings. This caused me difficulty at first, but the ability to eat a variety of nuts seemed to help offset my cravings which TOTALLY surprised me. Like Adkins, South Beach does eliminate whole grains those first two weeks.
SDB encourages eating foods in their natural form as much as possible, aka, clean eating so it is a very healthy way of eating.
Oh yeah, one more thing, while on Adkins, your body eliminates Keytones through saliva and urine. This phenomenon does not occur on SBD. I don't know if this is good or bad but the book and message boards I've seen makes a point of mentioning this.