I wrote this up while listening to an interview by Kevin Swanson with John Frame, on Two Kingdom Theology.
The Church and state are different. Church doesn't administer civil justice and the state does not administer sacraments or preach the Gospel. Everyone agrees with this. This is not two-kingdom theology.
Two kingdom theology adds to this that the state has to be, or should be, secular. Culture and state should operate on common grace wisdom unrelated to Christ's redemption, not God's revelation from Scripture. Separation of church and state is good and right. Christians should argue from natural law and reason in the public square, not appealing to the Bible.
This is where things get sticky. There IS wisdom given us unrelated to our redemption. Frame says we suppress this, condemning ourselves (Romans 1). But I'm not sure this covers ALL the skill or wisdom we have in the world. There is also Genesis 4:19-22 - cultural skill and knowledge (is this tainted by its Cain and Lamech context?). And there IS some level of separation and difference between the church and state. But to say they operate on fundamentally different paradigms, such that there are two different kingdoms is not helpful.
Where is the kingdom of God?
Kuyper says that there isn't one square inch that Christ doesn't claim as His. We are to do everything we do to God's glory, not just in our worship, but in our work, where Christ's kingdom is also present. Opposed to this, two-kingdom theology says that the kingdom of God is in the church worshiping on Sunday, not in the secular state.
Law and gospel
We are saved by grace apart from works of the law. This is basic. Luther went on to say that some Scriptures are law passages, and some Scriptures are gospel passages. Frame (and I) finds this problematic. Law passages can positively guide us; Gospel passages can smell like death to unbelievers. Obedience is good news to believers; the free offer of grace is repulsive to unbelievers. Two kingdom theology tends to overlay the sharp distinction between law and gospel, on their sharp distinction between state and church. To the state belongs the law; to the church belongs the gospel. This is way too simplistic.
In the end, we need to contribute to culture as Christians, and see that activity as kingdom work. Christ's kingdom shows itself throughout the world as leaven spreads in bread dough.
We do not have two kingdoms separated by law and gospel. We have one King, one kingdom. Granted, He does deal with different parts of His realm differently. But the way He structures His courts and counsels at the castle (the Church) are meant by design to do certain things in the rest of the realm (state and culture).