How do the president roles of chief executive and commander in chief differ

The commander in chief role deals only with the military, while the chief executive role is broader.

Explanation:

The commander in chief is the highest authority of the armed forces. In the case of the United States, a civilian, the President, is the Commander in Chief. Supposedly, this decision is meant to keep the military in check.

Chief executive is about the political branch in which the President is also the highest power. In the United States, we have 3 branches of power (The executive, judiciary, and legislature). The president is the highest authority of the executive power.