An assassination is the murder of a public figure. Assassinations are usually politically motivated. If someone kills your dog, that’s not an assassination, that’s just murder (unless your dog was running for mayor).
A murder is the unjust, illegal killing of someone. An assassination is a type of murder in which the victim is someone well known, usually in the world of politics. The killings of John F. Kennedy and Martin Luther King were assassinations: their purpose was to destabilize the government and hurt the civil rights movement, respectively. As assassination is murder plus politics.