Probably the biggest difference between Kline and Theonomists is their understanding of covenant theology. Kline emphasized the "different dispensations" or "administrations" of the one covenant of grace. Theonomists stress the continuity of the one covenant of grace.
Another difference is that Klineans put a great amount of stress upon the necessity and distinctions of the covenant of works and the covenant of grace. This is one reason why so many opposed the Federal Vision. Some Theonomist too often denied the covenant of works (or the merit based character of that covenant) in favor of mono-covenantalism, which usually ends up emphasizing the human end of the covenant obligations rather than the grace.
The biggest difference shows itself in their different approaches to society. Because Klineans hold so firmly to the Adam/Christ or COW/COG distinction (which is necessary), they too often view the present evil age as not worth redeeming. For them, this present age is a sinking ship, and the role of the gospel is to evacuate the elect into the Church and bring them to the new heavens and earth.
The Theonomists on the other hand view the present age more as in need of remodeling, purging out the corruptions of Adam and replacing it with the righteousness of God's kingdom. The role of the gospel then is to recruit people into the kingdom and transform the present evil age into the new heavens and earth over time.
I think you can see from this, why more Klineans favor Amil and Theonomists favor post-mil eschatology. They have different understandings as to the broader role and nature of the gospel and how the kingdom of God affects society before the return of Christ.
In the unhelpful rhetoric, Theonomists paint Klineans as pessimistic gnostics with no concern for the real world or real people. Klineans paint Theonomists as worldly political zealots who would rather take the world by the sword than through the gospel.
Granted, this is my own understanding of the dispute, and I'm painting with very broad brushes, and I' m open to correction. (No doubt may will offer correction!) I think rather than pick sides, these two can learn from each other.
