Hi ChaseWind,
This is an issue I struggle with for a few years and agree, on the surface it would appear they are different but I would ask you to consider the primary sources, the sources written by the framers of the First Confession and what they latter said about the Moral Law of God in the Second Confession. The Primitive Baptists (sometimes called Regular, Old Line, Old School, Hard Shell) and NCT Baptists do us a disservice when they post on the internet material that places emphasis on what they believe are the differences. It really just muddies the water. The First used a lot of non-theological language and the Second adopted almost completely the theological language of the Savoy and Westminster Confessions.
I will admit that some Baptists were antinomian and others were neonomian but most agreed with the 1689 on the Moral Law which is why it was signed and used by so many churches. It is anachronistic to view the First Confession as antinomian when the Second so clearly agrees with the third use of the Law. Dr. Renihan has written a short article explaining some of the details.
This quote from John Gill (1697 - 1771) might be helpful in explain the Moral Law, the Law of Christ and the Terror of the Law,
The law is in the hands of Christ as a rule of walk and conversation, directing believers how to conduct and behave themselves under his influence. The whole scripture, given by inspiration of God, is the standard of faith and practice, and the rule of both; the gospel-part of it is profitable for doctrine, and is the test of that; and the law-part of it respects duty, and points to that; wherefore to the law and to the testimony; if men speak not, and act not according to this word, it is because there is no light in them. (Isai. 8:20)Christ is king and lawgiver in his house and kingdom, the church and besides some positive commands which he has delivered out, there is a repetition of the law in the New Testament; a new edition of it, published under the authority and sanction of Christ; so that we are now under the law to him, (1 Cor. 9:21)and under new obligations to obey it, as held forth by him. And it is to be obeyed from love, in faith, and to the glory of God, without any sinister, selfish, mercenary ends and views.
It is to he obeyed from love to God and Christ; the end of the commandment is charity, or love; out of a pure heart, and of a good conscience, and faith unfeigned: (1 Tim. 1:5) not the terrors of the law, but the love of Christ constrains believers in him to yield a cheerful obedience to it: which they do through faith in him, depending on him for grace and strength to serve him in it. Of all men in the world none are under greater obligations to be careful to maintain good works than believers, and none so capable of performing them as they, and none so ready to do them; and in doing which they seek not themselves, but the glory of God; and which, as it should be, they make their chief end, as in civil things, so much more in religious duties; and when they have done all they can, and are assisted to do, they own they are but unprofitable servants; do not and cannot merit any thing at the hands of God, but expect eternal life and salvation as the free gift of God through Christ.
And now, true believers, who behold the law in the hand of Christ, and as fulfilled by him, delight in it, after the inward man; and though with the flesh they serve the law of sin, to the grief and distress of their souls, yet with the mind the law of God. (Rom. 7:22, 25)"
PS: Theological terms such as 'Moral Law' are used to sum up theological ideas contained in scripture. Like saying 'Holy Trinity' or eschatology, supralapsarianism or reprobation...they all have biblical meaning and used to convey biblical ideas.