So how is the C.A.R. system fundamentally different from a Weaver stance?
Well, one big difference is that, AFAIK, Weaver isn't taught right hand - left eye, and vice versa.
I sometimes try shooting "new isosceles", but after 35 years of shooting modified Weaver, it's the stance I find myself in when I "don't think, shoot". Of course Weaver was the rage (much due to Jeff Cooper) at the time I started shooting defensive pistols as a replacement to the isosceles that the "old-timers" used. Now isosceles is the replacement to the Weaver stance the "old-timers" use. Twenty years from now some hot shot shooter will use another modified Weaver, make it popular, and that will be the stance of choice.
I figure shoot whatever way gets your rounds where they need to go, quickly and efficiently. I don't know that I could get good with CAR as a standard stance, though I can see a value to it in a situation like sitting in your car and having to shoot a potential carjacker from either side. Otherwise it looks more like it's designed for operators (the real ones).