What is the difference between one's motive and one's intent to commit a crime, and which is the most important in our legal system?

A motive, in law, especially criminal law, is the cause that moves people to induce a certain action ... The law technically distinguishes between motive and intent. "Intent" in criminal law is synonymous with Mens rea, which means the mental state shows liability which is enforced by law as an element of a crime.