Did Malcolm X and Martin Luther King disagree about the goal or only the method?

They disagreed about both, but not in the same way throughout, and not for ever. When Malcolm X spoke for the Nation of Islam, he disagreed with King about the goal, separation instead of integration, and about the method, "by any means necessary" instead of nonviolence. After he left the Nation in 1964, some of that changed. It helps to keep the two periods apart.

In his Nation of Islam years, Malcolm X was its public face for 12 years, advocating Black empowerment and the separation of Black and white Americans, and criticizing King and the mainstream movement for its emphasis on nonviolence and integration. He rejected nonviolence, arguing that Black people should defend and advance themselves "by any means necessary". He called the 1963 March on Washington "the farce on Washington". The NAACP and other civil rights organizations, for their part, denounced him as an irresponsible extremist.

Then he changed course. In 1964 he transitioned to Sunni Islam. After the Hajj, he said that seeing Muslims of all colors interacting as equals led him to see Islam as a means by which racial problems could be overcome. In a letter from Mecca he wrote that his experiences with white people had convinced him to rearrange his thinking about race. He announced his willingness to work with civil rights leaders.

Some differences remained. He still argued that Black people should protect themselves if the government would not, and in "The Ballot or the Bullet" he told people to vote wisely but warned of taking up arms if equality was blocked. He was assassinated on 21 February 1965.