He throws it away because it's a grotesque trophy he never sought, given as a reward for a deed he did not consider heroic. All he'd done is killed a man in self-defense – and as he says later, as much as Tuco "deserved" that bullet between the eyes, and as much as the experience changed him and continued to change him, Hank did not want to become the kind of man who kept such things, as if they're something to be proud of. His focus in life should be on protecting people, not relishing in the taking of lives, even when those lives needed to be taken.
It was also a reminder of a moment he was frightened and, while not helpless, very out of control of his life. Someone tried to kill him and almost succeeded, and it shook him to his core. Throwing away the macabre trophy served the dual purpose of letting Hank live with the shooting not as a victory in some toxic macho sport (which is the way the cartel sees this crap, and Hank wants to be better than them, and knows he should be) but simply a necessary act that preserved his life, and a way to symbolically begin to put the trauma behind him.