How and why do Sunni and Shi'ite Muslims differ?

The main difference is their view on who Muhammad's successor should/should've been.

The people that believed that anyone with the right qualifications could take over insisted that Muhammad's father in law and friend, Abu Bakr, success him and became known as the Sunni.

The people who insisted that only someone blood related to Muhammad could take over, they wanted Muhammad's cousin and son-in-law, Ali ibn Abi Talib, to take over; and they became known as the Shi'ite.

They both follow the foundational practices and beliefs (Fice Pillars) of Islam though there are differences in the way they do some things and their views on the world. For example: Shi'ite Muslims complete all five daily ritual prayers but squeeze them into three sessions instead of five. Shi'ite Muslims also believe in the 'ten obligatory acts' in addition to the Five Pillars.

Hope this helps

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