We all know very well that the Hierarchs serve as the Covenant's heads of state, government, and religion, but we don't know how far that power extends. Furthermore, if public opinion turns against them, they could very easily be removed from power.
Here's a well-known example: Regret was able to secure his position as Hierarch by blackmailing the Prophet of Restraint with the knowledge that he had impregnated a San'Shyuum female even though he was on the Roll of Celibates. If Regret had released the information, the scandal would cause Restraint to be dismissed from his position. If the Prophets' rule was absolute, blackmail would have been impossible, as Restraint could simply have had Regret silenced.
Now let's look at the High Council: It could simply be a rubber-stamp organisation with no real power, an advisor committee, or essentially the Covenant's legislature. It seems that some sort of vote is required to bring the Hierarchs to power; Halo: Contact Harvest tells us that the Minister who would become the Prophet of Truth was involved in "compromise and coalition building," which implies that he had to secure his own power base by dealing out numerous favours and negotiating with other Council members so that if he was kicked out, so would they. He required support to stay at the top, and if he lost that support, he would face the Covenant eqivalent of a vote of no confidence, and be forced to resign. Contact Harvest, page 380 talks about "debate", and the Prophet of Restraint "argued" for greater funding towards gene therapy projects, so oratory is a necessary skill as well.
However, by 2552, Truth had apparently managed to gain a disproportionate amount of power over Regret and Mercy and the High Council. He was able to veto the Council's decision to have Thel 'Vadam publicly executed, although this may be a Hierarch's prerogative; the President of the United States can do something similar, or so I gather. Truth was also able to elevate the Jiralhanae into important positions in the Covenant, and was also able to enact the "Changing of the Guard", even though there would be extreme opposition from within the Hign Council's Sangheili camp.
So, in peace time, the Prophets are bound by constitutional conventions that limit their power. In Truth's case, he was probably able to gain emergency powers since they were at war at the time, presumably sometime in 2531, after the Covenant was soundly defeated at the Assault on the Apex.