Thinking Through Church and State
Some good thoughts on the differing purposes of Church and state from James Bannerman:
In the second place, the state and the Church are essentially distinct in regard to the primary objects for which they were instituted.
The state, or civil government, has been ordained by God for the purpose of promoting and securing, as its primary object, the outward order and good of human society; and that object it is its mission to accomplish wherever it is found, — whether in Christian or heathen lands. Without civil order or government in some shape or other, human society could not exist at all; and as the ordinance of God for all, its direct and immediate aim to aid the cause of humanity as such, without limitation or restriction to humanity as christianized.
On the other hand, the Church of Christ has been instituted by Him for the purpose of advancing and upholding the work of grace on the earth, being limited, in its primary object, to promoting the spiritual interests of the Christian community among which it is found. No doubt there are secondary objects, which both civil government on the one hand, and the Church on the other, are fitted and intended to subserve, in addition to those of a primary kind.
The state, as the ordinance of God, can never be absolved from its allegiance to Him, and can never be exempted from the duty of seeking to advance His glory and to promote His purposes of grace on the earth. And in like manner the Church, in addition to the objects of a spiritual kind which it seeks to accomplish, may be adapted, and is adapted, to further the mere temporal and social wellbeing of society.
But still the grand distinction cannot be overlooked, that marks out the primary objects of the Church and state respectively as separate, and not to be confounded. They are instituted for widely different ends. The one, as founded in nature, was meant primarily to subserve the temporal good of mankind; the other, as founded in grace, was designed primarily to advance their spiritual wellbeing. They may indirectly, and as a secondary duty, fulfill certain ends common to both; they may concur in contemplating certain objects together; but as they differ in their origin, so also they differ entirely in the primary and immediate purpose for which they are respectively established on the earth. (The Church of Christ, Vol. 1, 98-99)
Along similar lines, Guy Waters explains that although Christ reigns over all (Church and state), he does not reign over all in the same way or to the same effect. That is to say, there is a difference between the essential reign of Christ and the mediatorial reign:
Jesus' essential reign belongs to him as Second Person of the Godhead. This reign is unchanging and unchangeable. Jesus' mediatorial reign, however, belongs to him as the Incarnate Son of God, the God-man. This reign he assumed upon his exaltation. This reign is subject to change and may be said to increase or grow. This reign extends to the ends of the earth, but has the church as its particular focus…In other words, in acknowledging Jesus' essential reign, believers particularly confess Jesus, with the Father and the Holy Spirit, to be God over all. In acknowledging Jesus' mediatorial reign, believers particularly confess Jesus to be the risen Lord who purchased them by his own blood. (How Jesus Runs the Church, 33).
The books by Bannerman and Waters are both good. Start with Waters and move on to Bannerman if you want more. I commend them both.