When and why did the difference between the Ahmadiyya Muslim Jamaat and other Muslims arise?

There is no difference from Islam, only in the understanding of Islam. It came to light when Hazrat Mirza Ghulam Ahmad (as), the Promised Messiah, declared that Jesus (as) had died like all prophets, and that his expected return meant the coming of a man in his spirit.

The question needs to be put slightly differently. There is no difference between Islam and the Ahmadiyya Muslim Jamaat. Islam is the unchanging religion for all of humanity. Where differences exist, they lie in the understanding of Islam.

These differences came clearly to light when Hazrat Mirza Ghulam Ahmad (as), the Promised Messiah, announced what God had revealed to him. Jesus (as) was not the Son of God but a prophet. Like all prophets he fulfilled his mission and then died. None of them is still physically alive on earth today. The verses above say the same.

This raised a question. Muslims expect Jesus (as) to come in the last days. If he has died, who is to come? The answer was that a man would appear from among the Muslims, in the character and manner of Jesus (as). He bears his name as a title, because he is to do for Islam what Jesus (as) did for Judaism. Hazrat Mirza Ghulam Ahmad (as) declared that he was this Promised Messiah.

This matches what has always happened in history. Many religions expect the return of someone who lived long ago. Yet no such person has ever come back in person. The Jews, too, expected Elijah to return before the Messiah. Jesus (as) explained that this had been fulfilled in John the Baptist, that is, in another man of the same spirit.

In his attitude as well, the Promised Messiah resembles Jesus (as). Like him he placed love above retaliation and rejected spreading faith by the sword. It was over these questions that opinions divided, not over the religion itself.