A mane uncuted and a tail unwrapped is a real danger because they can catch themselves to the mallet or to the reins while the rider is trying to control the horse.
Therefore, the mane is shaved and the tail is wrapped to prevent danger. Thus, it allows to avoid the crual practice of the amputation, where the horse ended with a short stump.
Mane : https://www.youtube.com/watch?v=HkUIv7U2B9U&list=UUyH4tr5xhMzRvB8G4rWX4mw
Tail : https://www.youtube.com/watch?v=MgxckkKq7nU&list=UUyH4tr5xhMzRvB8G4rWX4mw