perhaps the difference lies within a legal term. A killer can be anyone that takes the life of anything, but a murderer is a person who took the life of a person criminally. There are also instances where a person takes the life of another person and both of these labels don't apply, for example, someone who kills in self defense or in defense of others (police officers, soldiers)

A killer can be free from judicial prosecution, the act of a killer (killing) does not always necessarily involves contradiction with law, for example, a tiger killing its prey, a soldier killing other opposing soldiers or terrorists. But "murderer" highly signals that its exclusively an act which is against the law, a crime, which is why its defined in a criminal manner, in a judicial way, murder -> murderer. The word murder/murderer highlights the implication of criminal act in it, a killing which is criminal in its essence.

A killer is someone who kills, whether by intent or not. A murderer is someone who kills with a motive in mind. In everyday speech, the subtle difference can be overlooked but in law, it can mean a death sentence or not. Legal definitions and everyday ones can be totally different!

All of the answers people gave above are correct, but for most English speakers these two words are interchangeable. You could use either word in most circumstances and they would have the same meaning.

Technically, there is a difference! To "kill" is to cause death. One ANIMAL will "kill" another for food. You do not hear English speakers say that a tiger "murdered" a cat. No one says a cat "murdered" a bird. The word "murder" is associated with deliberate, often premeditated killing of one human against another HUMAN. However, English speakers in everyday speech will often say, "he was a cop killer" or "he was a serial killer". Hope this helps