It's all about the caliphs, and goes back a long way. After the death of Mohammed, one group chose a new caliph for his ability, the other group a different one for his relationship (cousin) to Mohammed.

The latter group eventually lost, and generally since then the Shias have avoided a link between religion and the state, since political power was held to lack legitimacy. This was held right up to the time of the Iranian revolution. It's a sobering thought that a bit more flexibility by the west over the Shah's dictatorship could have avoided the whole horrible mess since.