On CT versus Baptist CT, I think it's largely a difference of opinion about what is essential versus accidental in the covenant of grace. CT'ers see "to you and your seed" as an essential part of the gospel promise, as the way God always works with his people. Baptist CT'ers view the inclusion of children in the covenant as typological, indicating the line through which Messiah would come. So, once Messiah has come, there is no longer any need for the type. Therefore, baptism actually signifies less than circumcision did, since baptism symbolizes purely spiritual realities, whereas circumcision had ethnic and social meanings that are no longer applicable. This is the basic argument of Paul Jewett's Infant Baptism and the Covenant of Grace.
This is the heart of the issue. In both schemas, the Covenant of Grace is in Christ. Nobody ever has nor ever will be saved by the Covenant of Grace who is not united to Christ by faith. This is what Paul labors about Abraham in Romans 4. It's also what the author of Hebrews notes in Hebrews 11. The essential character of the Covenant is faith that God's promises are "Yes" and "Amen". He has promised to save men by grace. Men who trust the Promises of God are truly in Covenant with Him by His electing grace. The Advent of Christ changes not the fundamental character but is the final revelation of that Covenant and also the full guarantee of it. Former administrations were instructional in the sense that they were instituted for the purpose that they might be replaced by the antitype that they typified. The author of Hebrews labors this point at length that God instituted an imperfect system that it might be fulfilled in Christ and that those who approached God by those OT sacrifices ultimately did so because of the reality that they signified.
Christ's coming then is not the advent of a new God and a new way of salvation but the full revelation thereof. The Son of God comes to be a Servant Who perfectly does the Father's will and fulfills every stipulation of Covenant obedience. He is the perfect Messiah because He came to obey and obeyed perfectly. He is the perfect Priest, which the imperfect Aaronic priesthood typified, and His sacrifice is perfect and once-for-all.
That all said, then, as Charlie notes, the real question is whether the Covenant signs of the OT served merely as some sort of biological or national purpose disconnected from the essential character of the Covenant of Grace. Baptists tend to see the sign of circumcision as a national sign that biologically ensures Christ will come but not really connected to the Covenant of Grace in an essential fashion.
Because the Covenant of Grace has always and now consists of the elect alone, the Baptist insists that anything that participates in that perfect Covenant has to, itself, be perfect. An elect person who was circumcised participated in the covenant but so did a reprobate person who was circumcised. Thus, they conclude that because circumcision was applied indiscriminately of whether or not the person was elect, the perfect nature of the CoG excludes any notion that circumcision could be somehow essentially tied to the CoG. The fact that Abraham has Ishmael circumcised provides the appropriate out in this case. Because Abraham is given a land and physical promise, circumcision sort of becomes tied more to a physical and national aspect of a promise made to Abraham that does not require that it be thought of any way of entrance into the perfect CoG that a reprobate Jew might defile by his reprobate-ness. The Old Covenant as well, under Moses, is seen to have this character by the Baptist because the whole nation, elect or reprobate, participates in the "covenant life" of its administration. The perfect Covenant of Grace, then, cannot be seen as being coextensive with the Old Covenant because, again, there is this "defilement" by reprobate Jews who are never really united to Christ by faith.
When the New Covenant comes, then, the Baptist sees Jer 31 and Heb 8 as the idea that God is going to no longer have a visible Covenant on earth. The New Covenant is the Covenant of Grace in Christ. It is with the elect of Christ alone. There is no longer any visible Covenant, per se, in Baptist theology. Why? Because of the admission that, even with the best professors and lives that indicate, there may be false brethren in the Churches. Thus, the thread continues where an elect person who is baptized participates in the New Covenant but a reprobate person who is baptized does not. New Covenant = Covenant of Grace = perfect. The reprobate person cannot in any way participate because that would make an imperfection in the NC. Hence, baptism itself (like circumcision) is not of the essence of the CoG. Where circumcision was not of the essence of the CoG, it did serve (by Baptist thinking) to make one a member of the OC but, again, that was not coextensive with the CoG. Now baptism is not of the essence of the CoG and does not make one a member of the NC but does make one a member of the local Church.
I'm getting an Excedrin Headache trying to get to my point.
At the end of the day, the Baptist treats the CoG as so ideal and perfect as to leave it out of the reach of any historical administration. Its composition is and always has been in the mind of God alone. He knows His elect and saves them but neither circumcision nor baptism has ever been a means by which a man could positively state - "I am in the Coveant of Grace". In the OC, circumcision served to let a man know he was in the OC but not necessarily in the CoG. In the NC, baptism serves as a testimony that a man claims to believe but it serves no function to assure anyone that he is truly in the NC.
The Reformed position does not deny the inscrutable character of God with respect to His knowledge of the Elect of God but sees God's redemptive signs as methods of divine condescension. God does not leave knowledge of the elect beyond history but is pleased to give Sacraments to His Church by means of which men can participate in historical acts and, by the eyes of faith, be drawn to spiritual realities that exist beyond history.
That's the simplest way I can explain it at the moment.