What is the difference between the head of state and the head of government?
One is a religious figure, the other is a secular figure.
Historically, going back thousands of years, the head of state was a religious figure, and the head of government was a secular figure. In authoritarian regimes, the two are often unified, but in a liberal society the two are as separate as religion and government.
I myself am highly secular. I think highly secular people have trouble understanding why the head of state should exist, just as we have trouble understanding why religion needs to exist.
For those who are open to the religious element, the head of state represents the mystical and mysterious union of the people, the spirit of the Volk made incarnate in human flesh.
It is worth noting that in those European nations that still have monarchies, the people report higher levels of trust in government. It is possible that the secular republics, who have overthrown their monarchies, make a mistake by have the head of state serve for such short terms, as if they were ordinary politicians. To the extent that the people want a symbol of unity, such leaders should probably serve for longer terms.
And then of course, there is the United States Of America. I sometimes jokingly say there are 3 types of government in the world:
democracies, with separate head of state and head of government
dictatorships
The United States Of America
People in the USA suffer a bit of schizophrenia when they look at their President. When they like the President, they see the President as a head of state, a monarch who should not be criticized. But when they don't like the President, they see the President as a head of government, a lowly politician to be torn apart on comedy shows.