What gave rise to a fundamental division in the umma and in Muslim theology?

Answer 1

There really was no 'fundamental division in theology". Silly as it may sound in view of IS atrocities and the Sunni / Shia fights in Iraq and Syria, the major difference between the two beliefs is purely the result of a dispute over leadership in the 7th (!) century AD. The Sunnis think that a guy named Abu Bakr and his family was the prophet Muhammad's rightful successor, while the Shiites thought that is was Muhammad's son-in-law Ali.

The theological differences are few and all very minor. Until the end of the 20th century, Shiites and Sunnites generally lived peacefully together and intermarriage was common.

Today's division is mostly the result of an ongoing struggle for political dominance of the Middle East between Sunni Saudi Arabia and Shiite Iran - plus in the past few years, Sunni Turkey that hankers for a return of some of its former Ottoman Empire greatness.

Answer 2

General Answer
It happens with Islam as it happens with most religions of the world. Strong personalities and differing cultural needs led to various kinds of divisions. These divisions are sometimes interpreted as breaks with orthodox teaching, and so various sects take root. The divisions can form around issues of scriptural interpretation, the application of various teachings, or the nature of authority in the religion and how that is going to be passed down.

Specific to Islam
There are numerous splits in Islam, these are probably the most historically important.

The Sunni-Shiite Split
The main reason for the split here was: Disagreements over Succession after Muhammad's death.

The Muslim community was united while Muhammad was the leader of this community. Most Muslims hold, however, that he never specifically chose a successor to his leadership. There was a minority in the community that supported the candidacy of 'Ali, the Prophet's son-in-law, this political faction became known as the "Supporters of 'Ali" which in Arabic is Shiat 'Ali (where the modern term "Shiite" comes from). They derived their support from specific hadiths and events that they claimed showed that God had revealed to Muhammad that 'Ali would succeed him. The majority of Muslims held that these hadiths and events showed nothing more than that 'Ali was very pious, something they did not deny. Therefore they gave power to the man who was Muhammad's second-in-command and father-in-law Abu Bakr. This majority were called the People of the Customs [of the Prophet] which in Arabic is Ahl Sunna (from where the modern term "Sunni" comes from.)

The Sunni & Shiite-Kharijite/Ibadi Split
The main reason for the split here was: Impurity of Leadership.

The Kharijites were dominant in Egypt at the same time that the Shiites and Sunnis saw their first major confrontation: the Fitna Al-Kubra. At this point, the Kharijites fought along with the Shiites against the Sunnis and were able to press the Sunni leader, Mu'awiya, to the precipice of defeat. However, Mu'awiya asked to treat with 'Ali, the Fourth Rightly-Guided Caliph and leader of the Shiites. 'Ali's accepting of this invitation and the trickery that Mu'awiya played on him during the meeting deeply angered the Kharijites who believed that 'Ali was too easily deceived to be a true leader of God. They abandoned the Shiites and became a third party to the conflict. This made it possible for Mu'awiya to emerge victorious and conquer the Shiites. He then turned his attention to Kharijites and routed them. Kharijites disappeared completely a few centuries later, but they were succeeded by a less puritanical subgroup called the Ibadis who form the majority in Oman.

The Sunni-Mu'tazilite Split
The main reason for the split here was: Whether Islamic Law should be viewed through the lens of rational thought or purely Divine Edict.

The Mu'tazilites developed as a school in Baghdad that taught that a rational outlook on the Qur'anic traditions and the discussion of God, postulating a creation of the Qur'an, actual time-bound actions of God, and momentary attributes. In contrast, mainstream Sunni Islam at the time endorsed the idea that everything about God was co-eternal with Him (including any speech he made or books he revealed). While Mu'tazilites had power in Baghdad for roughly 200 years, they slowly waned and were "outvoted" by traditional Sunni theology.

The Sunni & Shiite-"Modern Prophets Islam" Split
The main reason for the split here was: Whether a Prophet could arise after the Death of Mohammed.

There have been a number of different Islamic Sects (like the Druze, the Baha'i, and the Ahmadiyya) who disavow the Muslim claim that Mohammed was the final prophet and the seal of the prophet. This betrays one of the fundamental claims of Islam. In order to avoid scrutiny and persecution about this, the Druze and Baha'i now self-advocate as different non-Islamic religions. The Ahmadiyya, however, still claim to be Muslim and are subject to violence due to their "apparent heresy".
The main reason for the split here was: Impurity of Leadership.

The Kharijites were dominant in Egypt at the same time that the Shiites and Sunnis saw their first major confrontation: the Fitna Al-Kubra. At this point, the Kharijites fought along with the Shiites against the Sunnis and were able to press the Sunni leader, Mu'awiya, to the precipice of defeat. However, Mu'awiya asked to treat with 'Ali, the Fourth Rightly-Guided Caliph and leader of the Shiites. 'Ali's accepting of this invitation and the trickery that Mu'awiya played on him during the meeting deeply angered the Kharijites who believed that 'Ali was too easily deceived to be a true leader of God. They abandoned the Shiites and became a third party to the conflict. This made it possible for Mu'awiya to emerge victorious and conquer the Shiites. He then turned his attention to Kharijites and routed them. Kharijites disappeared completely a few centuries later, but they were succeeded by a less puritanical subgroup called the Ibadis who form the majority in Oman.

The Sunni-Mu'tazilite Split
The main reason for the split here was: Whether Islamic Law should be viewed through the lens of rational thought or purely Divine Edict.

The Mu'tazilites developed as a school in Baghdad that taught that a rational outlook on the Qur'anic traditions and the discussion of God, postulating a creation of the Qur'an, actual time-bound actions of God, and momentary attributes. In contrast, mainstream Sunni Islam at the time endorsed the idea that everything about God was co-eternal with Him (including any speech he made or books he revealed). While Mu'tazilites had power in Baghdad for roughly 200 years, they slowly waned and were "outvoted" by traditional Sunni theology.

The Sunni & Shiite-"Modern Prophets Islam" Split
The main reason for the split here was: Whether a Prophet could arise after the Death of Mohammed.

There have been a number of different Islamic Sects (like the Druze, the Baha'i, and the Ahmadiyya) who disavow the Muslim claim that Mohammed was the final prophet and the seal of the prophet. This betrays one of the fundamental claims of Islam. In order to avoid scrutiny and persecution about this, the Druze and Baha'i now self-advocate as different non-Islamic religions. The Ahmadiyya, however, still claim to be Muslim and are subject to violence due to their "apparent heresy".