I've tried using a lunge whip, to "whippy". The advantage of the carrot stick is that it is stiff. It allows the use of steady pressure on the horse which is not possible with a flexible stick.
I have also found by testing on my wife (just kidding, my hand actually :) ) that used with the same force a flexible whip causes much more pain than I can achieve with a carrot stick. I assume that CAs stick has similar characteristics as he probably got the idea when he was training under the Parelli system. I'd just choose the cheapest.
Incidentally I have seen cheap copies of the carrot stick by other manufacturers and considered buying them, but they were just to flexible.
Believe it or not, that is why I prefer to use a dressage whip or a buggy whip when lunging. IF, and I say IF I need a hrose to move off or away from me NOW! Then I don't feel that the carrot stick has the umph, nor can I manipulate it with the speed I need. And due to it's weight and clumsiness, if I am working up close to a horse, having them circle me real close, and I need to get a shoulder off me pronto, I can't get the darn carrot stick to do much at close quarters.
A dressage whip just has to be vibrated in the air real quick and the hrose will move away immediately. I rarely ever actually hit a horse. Mostly either vibrate to make a noise, or just touch them lightly with the soft tip.
I see folks with carrot sticks swinging and swinging and swinging the rope, and too many times the horse makes no change. So, the biggesdt they can get with the carrot stick gets no response, so the hrose ends up needing MORE.
However, I have seen Pat himself (in video) working with a horse at liberty with a carrot stick and it was a joy to watch.