What is the difference between the New Covenant and the covenant God made at the Exodus?
The New Covenant is made specifically with the house of Israel and Judah, unlike the Exodus covenant God made with their fathers when He brought them out of Egypt, which they repeatedly broke. In the New Covenant, God will put His law in their inward parts and write it on their hearts, becoming their God and they His people.
Les explains that Jeremiah 31:31-32 shows the New Covenant is made with the house of Israel and the house of Judah, and no one but. It is not according to the covenant God made with their fathers in the day He took them by the hand to bring them out of Egypt, which covenant they broke over and over again, although He was a husband to them. In the New Covenant, God will put His law in their inward parts and write it in their hearts; He will be their God, and they shall be His people. Les teaches this will not be Israel's choice. When Christ returns and sets up His Kingdom, the remnant of Jews kept through the tribulation period will see Him coming in the clouds of glory and believe in a moment. The nation will be born in a moment, or a day, spiritually, and then the New Covenant becomes a reality. He also notes that believers in the Age of Grace are not in a covenant relationship; they are members of the Body of Christ, though they enjoy the benefits of the work of the cross, the power of His resurrection, the coming of the Holy Spirit, and other blessings.
Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: