Ah the classic battle between good and evil - The Batman vs. The Joker. The Batman is the Joker's greatest arch nemesis and vice versa. Both have been in a constant struggle over the crime infested city of Gotham. If the Batman could kill the Joker, everything would be over. The Batman would have no more villains terrorizing Gotham's streets. Good ultimately triumphs over evil and Gotham's people are safe. If the Joker kills the Batman then there would be nothing left standing in his way of claiming Gotham for himself.
It's not as simple as it seems. Looking at both of their characters, we find many similarities. Both have suffered from a traumatic experience drastically shaping their identities forever. Both are very resourceful and smart. Both are ruthless in accomplishing their goals. There are however some differences that serve as advantages and disadvantages to both characters. The Batman will never kill the Joker let alone break his own moral code because it is what defines him. The Joker has no morals and thus does what he wants. The Batman let his trauma shape him into a vigilante in order to prevent it from happening to anyone else again. The Joker uses his trauma as proof to proclaim that because there is injustice, justice is meaningless. In simpler terms, life is all a big joke. The advantage to this is that the Joker has far more capabilities and opportunities to commit his crimes against the Dark Knight. The Batman has many friends which leads to danger as he is emotionally vulnerable if they were to be killed. The Joker has no friends which is also disadvantageous because there is no one to save him when he at his weakest.
Why won't the Batman and the Joker kill each other? My theory is that the Batman won't kill the Joker because he has a high conduct of justice. Killing a criminal without a fair trial would be murder and the Batman is no murderer. The Joker on the other hand could kill the Batman and Gotham would be his for the taking but that's too easy. The Joker wants to break the Batman's psyche by creating another trauma worse than that of his parents. The Joker wants the Batman to finally snap and kill him because then, the Joker's ideology would triumph. However, once the Batman commits an evil act and his moral code is broken then and only then will the Joker land the killing blow. In the meantime, the Joker continues killing innocent people to provoke the Batman to evil.
It's a never ending battle between the duo. Nevertheless, both will always continue to inflict excruciating pain on one another until they live to fight another day.