So there were many different conditions to try it out on.My experiences so far (and remember this was only my first and second days attempting this, and of course I wasn't laying it 90 degrees over - I was just trying to push-pull with my normal 60-70 degree turns):
It seemed really easy to learn because it is confidence inspiring. The first time I tried it I felt like I was just as in control of the turn as I normally am, which is atypical of learning new methods; usually my first time with a new method I feel worse for at least a little while. After a few runs I already felt more in control.
Compared to my normal turns, I felt considerably less G-force at the end of each turn. Not just 5% less or 10% less, but more like the difference between "I feel like I'm squatting 500 pounds and my quadraceps are about to explode from the strain" to "it's quite easy." Usually at the end of my turn, I am compressed with so much pressure I don't have the strength to do a whole lot. With push-pull, the pressure was so reduced I felt like I still had many options open. Maybe this is what Jacques+Patrice mean by being the "commander" of your board?
Push-pull felt superior on fresh groom, hardpack, icy-ish stuff, and even lumpy crud that's left over after skiiers and boarders trash a hill all day long. I'm not a good enough rider on real ice to be able to make a judgement on that.
Push-pull didn't work at all for me in the slush. Actually, it wasn't quite slush - it was a thick layer of chunks of hard snow (I think some people call them death cookies?), but I've found that that kind of snow rides like slush. In any case, when I started to pull at the end of the turn, my board unflexed and stopped carving an arc. Instead, it went straight downhill and I washed out. This happened many times, both heelside and toeside. Perhaps a softer board would have helped? I was riding a fairly stiff Donek Freecarve I.
Push-pull let me carve at lower speeds than I am used to. I'm not sure why.
Push-pull made edge transitions fast and effortless.
I'm not sure why they say push-pull turns need a very good physical condition (strong thighs and buttocks) on their page - it seemed much less demanding than my normal bomber-style turns.
I was riding with a rotated heelside turn, but not a rotated toeside turn. I was attempting to do toesides in the normal basic position (chest facing same direction as my boots). It seemed to work just fine...
Overall, I was expecting to find that push-pull turns would be better in some conditions and worse in others, but in general, I felt that as long as the snow was hard enough to keep the board flexed, it was mostly just better all the time. More surpisingly, I don't even have any more questions after having tried it - it seemed very intuitive - I'm just going to go out and practice some more.