The new DVD does a very good job of describing the GGA method in considerable detail. It's very well done, very professional. It incorporates their current training methods.
What follows is strictly my opinion based on my experience learning SA mostly from DVDs.
The current GGA and NG swings are very similar. When I look at Ken Martin or Todd Graves swinging, they are about the same. The differences are no more than you would see between any two individuals who use or teach the same method. Even Todd and Tim Graves do not swing exactly alike. Instruction-wise, NG could use a little more detail. Conversely, GGA is almost over-detailed. NG wants you to feel it and figure it out. GGA wants you to check and meet numerous positions and moves. Their instructions are valid but at some point I would probably have had a hard time getting it to the course. OTOH I often felt NG instruction was lacking some key points. GGA is from the school of thought (which has adherents here) that before you golf, you should spend a LOT of time learning setups, positions, and movements. NG I think is more like "here it is, feel it, try it, and go play some golf". I like that approach better but I do appreciate the detail of GGA now that I've played NG and have sought out specific fixes.
The methods differ in the prescribed hold of the club, with GGA teaching overlapping and NG teaching 10 finger. With apologies to Todd Graves, I think that is minor. What I do think is a major difference, however, is the interpretation of the lower body movement at the beginning of the downswing. I think that for either swing, the lead hip (or leg, knee, or whatever you focus on) does move at the beginning. NG teaches that the movement happens naturally in anticipation of and reaction to what is happening up top. But GGA emphasizes the lower move as something to be learned and cultivated. This might well be a personal idiosyncrasy of the instructors involved and what they feel during the swing. I know Moe said his leg/knee/whatever started first. But he also said he played on top of his legs. What I'm saying is that the lower starts first, but that doesn't mean you need to make it happen. Some might, but some might get screwed up by it.
Personally I find it very hard to consciously move the hip to start the downswing, because it's hard to keep it from taking over and throwing the swing off plane. What they are suppose to do is start, then they slow down and nearly stop till the swing passes by. So I prefer the NG instruction of swinging the upper body and letting the lower body react accordingly. It works much better for me. (And is the source of one of my most persistent problems, hence my fixation on it.) Overall, I think this distinction is the major difference between the two methods.