The carrot stick is not just a buggy whip with a different name. Whips are thin and flexible, they typically have a lash of some sort at the end, and they are made to sting if you need to crack a horse with them. The carrot stick is thick and quite heavy. The string on the end is made of yacht rope and is fairly thick so it cannot lash or sting a horse. You cannot flick the carrot stick at the horse and "pop" them like you can with a whip. Also the carrot stick doesn't make quite the whizzing or whooshing noise that a whip does when you move it through the air. The carrot stick is not intended to be used like a whip either.
I own carrot sticks, kid sticks (smaller version of the carrot stick), traditional dressage whips of different weights and lengths, a traditional driving whip, and a traditional lunge whip. They are all different with different uses and properties.
To say a carrot stick is the same as a buggy whip is like saying a Corvette and a Chevy 4x4 diesel are the same thing since they both have engines, steering wheels, and tires.......
Some properties may be the same, but other properties are different. There are a thousand different types of toilet paper with different properties, depending on what you prefer. To say one is better than the other is kind of a waste of time. Depends on what you like, and how you use it, as to which brand and style you end up buying.
When you use a traditional whip, it is too easy to lose your patients and "flick" the horse, or "pop" him when he doesn't respond. Or it can be done accidentally as well. A carrot stick is far more difficult to do this with. The weight and size of it, makes it difficult to manuever through the air quick enough to pop the horse with it and sting him. Sure you could abuse a horse with a carrot stick, but under normal applications, the size and weight of it makes it more difficult to be used as a snappy little discipline tool. You tend to use it more as an extention of your arm because it feels more like your arm than a thin little whip does.