a whip is for a correction. A whip is light and fast in his movement, so you give a very quick sign.
If you keep a whip in front of you and you move your hand a little the whip will immidiately follow your hand and it will go up and down. It moves in a fast way.
A carrot stick is for define things, to support your bodylanguage.
If you keept the carrot stick in front of you and you move your hand a little, the carrotstick will move slowly, just slightly a bit slower than you hand was moving. (a bit of a delay if you can say it that way in english.)
So you can not be fast which make it better to follow for the horse, so he can understand better what you ask.
It depents ofcourse which idea you have when you work with it. positive of negative reinforcement and so on.
i believe that the story of Parelli is that he called it a carrot stick to remind people that they have use it in a friendly way.
carrot is in the mind of people food for a horse as a reward, so it is friendly....
and then colours it orange and the carrot stick is born
I've seen videos of Pat using the carrot stick string to strike the horse to punish. He calls it a "spank." The incident I remember was to move the horse on a circle. The horse stood for one strike on the ground, then a second strike on the ground but closer to the horse, then his "spank," which was pretty hard. I could not do that to my horses.
As I commit more and more fully to +R, positive reward work, I find carrying anything that has a potential to threaten or hurt to be counterproductive. I don't want the horse thinking about threat.
In fact, because I use targeting so much, and that is an object on a old skinny dustpan handle, I do not want the horse associating the target with a whip or stick.
I think a lot of body language is in fact threat language in horse terms, what they use in the herd with each other - pressure. There are of course some directly positive gestures and invitation movements that aren't.
But if I am going to use strictly positive reward and capture a behavior and mark it with a sharp sound that is all I want the horse to experience - nothing negative or potentially negative.
I keep a stick and string, and in fact an old collection of various whips, but find so little use for them they are just relics, antiques, now.
I noticed that when I carried one while I was doing positive behavioral work it also distracted me from the task and goals of true positive work. Of course I have twenty years of pressure work to break the habit of. LOL