Let's start with the definition of "murder" as a verb: "to kill (someone) unlawfully and with premeditation." Your question should probably be "what is the difference between a murderer and a killer?". A murderer (noun) is someone who murders (verb) or commits a murder (noun). A killer (noun) is a someone or something who kills (verb). Note that a killer can be a thing (such as a disease or natural phenomenon such as a storm or earthquake). A murderer is almost always a human because only humans can kill with evil intent.

In real-life usage, there is overlap. According to strict use of the precise definitions, "to kill" is a broad term meaning to take life, while "to murder" is a legal term meaning to take life intentionally and illegally. Thus, an executioner who kills a criminal who has received a death sentence is a killer, but not a murderer. The executioner is acting legally. A soldier who kills another soldier in a war is a killer, but not a murderer. The soldier is acting legally. A person who kills another person by accident, or though carelessness, might not be a murderer. He might be guilty of some lesser offense like "manslaughter" or "negligent homicide."

All murderers are killers but not all killers are murderers.