The relationship between the church and the state has been a diverse one for a lot of different reasons. Generally speaking there have been four ways the church and the state have related to one another.
Erastinianism is the view that the state has authority over church and controls the church. It began when Erastus suggested that the state should execute the church’s sentence of excommunication and can be seen in some places even today where there are state churches and where the state receives taxes and then gives them to the church or owns the church property and pays the clergy. In the early days of the United States a national church was rejected but states had certain denominations that they supported. For example, the state church of Virginia was the Anglican church but the state church of Massachusetts was the Congregational church. To this day the British monarch is considered the chief governor in the church.
The opposite approach to Erastinianism is, of course, a theocracy, where the church controls the state. So, for example, when Pope Leo III crowned Charlemagne to be the Roman Emperor in 800 it sure looked like the church was in control of the state. Other manifestations of this were the Crusades wherein the state waged religious wars (Full disclosure: The Crusades are a complex subject and the last sentence should not be read as an indictment against the church. The Crusades were a response to Muslim aggression. For more on this see Rodney Stark’s God’s Battalions: The Case for the Crusades). Other manifestations of this can be seen by those who suggest that the United States is/was a Christian nation or that Christian leaders should be elected so as to institute Christian rules and principles.
The church is to be busily and faithfully doing its job and state is to be busily and faithfully doing its job.
Hearkening back further in history, Constantinianism, named after, of course, Constantine. The genius of Constantine’s approach and interaction with religious organizations was that he did not forbid or outlaw one and not others but rather favored one and directed the state’s resources to create an environment where the church he supported would flourish and other fade away. In his case, after his conversion, he favored the Christian church. In return for such favor, the church accommodated the state in order to retain its favor (for more on this see Peter Leithart’s Defending Constantine).
Finally, the approach that promoted the most separation was the approach that insists that the church and the state exist separately and have different functions to fulfill. Our Westminster Confession of Faith gets at this when it notes that:
Civil magistrates may not assume to themselves the administration of the Word and sacraments; or the power of the keys of the kingdom of heaven or, in the least, interfere in matters of faith (WCF 23.3).
To be sure, there are different manifestations and evolutions of each position and, to be sure, there is certainly historical overlap on some of these. Paul’s teaching in Romans 13, though, seems to look something like what is advocated by the fourth position referenced above. The church is to be busily and faithfully doing its job and state is to be busily and faithfully doing its job. There is to be a recognition that though the jobs are different they are both needed and they are both ordained and ministries of God. The church should work for the good of the state because the good of the state is for the good of the church all the while being very careful not to turn the state into an arm of the church.
Hopefully this historical and theological sketch is helpful as we think through our relationship with the state.