The atlatl style shown is a spur-less atlatl (or “split and wedge”). Instead of a spur, a cord is stretched tight in a Y-shaped split at the end of the atlatl.
The darts used would have to have their butt ends amended. Instead of a cup or cone shape in the rear of the dart, the darts would need a shallow groove cut across the end to accomodate the stretched string. A groove too deep would hamper a smooth release of the dart butt from the stretched string during the throwing motion, much the same way a too long and narrow and sharply pointed spur digs into the inner wall of the dart butt during the throwing motion.
The advantage of the stretched string method is that the dart butt could be cut at a node leaving a solid dart butt less likely to be damaged over time, as is the case with spur-and-cone dart butts. The second advantage would be the ease of manufacture as opposed to the work that goes into carving or building up a good spur.
I can think of no real disadvantage unless we are talking about atlatls with dart rests. The purpose of a dart rest is that the dart rest holds the dart allowing the atlatlist to use all his or her fingers to control the atlatl during the throwing motion. I think that a deeper groove would have to be cut into the dart butt to prevent the dart from sliding forward and off the string while the atlatlist holds the atlatl and dart ready for the throw. As stated earlier, a deeper groove in the dart butt may cause the dart to get hung up on the string during the throwing motion. That said, I am curious enough to give the thing a try.
Mike Richardson, the author, is a well respected atlatl and dart researcher. He wrote a terrific Masters Thesis on the subject of the atlatl and dart.