Now, after his death, the Muslim society was divided between two religio-political ideas.
One held that Muhammad cannot be succeeded in his capacity as a spiritual leader, and
his successors should only be political leaders of the Islamic state. On the other hand, a
minority reasoned that Muhammad, after 23 years of working on creating a society based
on a belief, would not just die leaving its fate to the uncertain future. This latter group argued
that the Islamic society needs a spiritual leader, as well as political one, and it will be better to keep both roles within the same person, and as such, one supposedly sanctioned as a spiritual authority by Muhammad himself. This latter group became the Shi'a, while the former (with some compromise, the more "secular" group, or those promoting a separation of "church" and state) became the Sunnis.
...It of course gets complicated, but the simple answer to the question of "what is the difference between the Shi'a and the Sunni", is "the Shi'a believe that a society needs both a spiritual and political leader, while Sunnis consider the leader of the Islamic society is only a political leader".
http://www.vishistorica.com/brain/archives/000858.php

2.  Groups with extremist beliefs have emerged from both sides.
Among those who claim to be Sunni Muslims are the Qadianies,
who believe that a person by the name of Mirza Ghulam Ahmed
appeared in the Indo-Pak subcontinent over a hundred years ago,
and that he was a prophet of Allah who received divine revelation.
Among the Shi'a there are the Abadiyyahs, who believe that 'Ali was
partly divine; the 'Alawies, who consider 'Ali virtually a prophet; and
the Druze, who consider an 11th-century descendant of 'Ali, al-Hakim,
to have been the embodiment of God. All groups that hold such views
are diametrically opposed to the agreed-upon fundamentals of Islam and
are not considered within the fold of Islam by the mainstream Shi'as and
Sunnis who constitute more than 90% of those who claim to be Muslim.