>>81618459 (OP)President is the head of state, equivalent to the king. Generally he's mostly a figurehead.
Prime Minister is the head of government. He is the executive, and does the actual decision-making.
America is an anomaly, because their President also takes the role of Prime Minister, and is both the head of state and head of government, similar to how it is in absolute monarchies where the king runs the show (note that most absolute monarchies do have deliberative assemblies, before you squeal about congress & the senate)