A key difference between the old covenant and the new is horizontal. In the old covenant, the elect/redeemed/remnant/spiritually circumcised are a subset of the covenant community/physically circumcised. In the new covenant the two are the same--by definition one is a member of the new covenant who is elect/redeemed/spiritually circumcised. Entrance is not based on birth but new birth, marked by baptism representing life from death.
Even after I became convinced of this from Jeremiah 31/Hebrews 8, 10, it was a class on 1-3 John taught by D.A. Carson at RTS-Orlando that helped me to see that the change from old to new covenant was not only horizontal in terms of membership, but also vertical in terms of structure.
In his excellent 1990 essay, "Evangelicals, Ecumenism and the Church," Carson explains:
In the sixth century B.C. the prophet Jeremiah, speaking for the LORD, foresees a time when people will no longer repeat the proverb, "The fathers have eaten sour grapes, and the children‘s teeth are set on edge" (Jeremiah 31:30). The history of Israel under the Mosaic covenant has been characterized by the outworking of this proverb. The covenantal structure was profoundly racial and tribal. Designated leaders prophets, priests, king, and occasionally other leaders such as the seventy elders or Bezaleel were endued with the Spirit, and spoke for God to the people and for the people to God (cf. Exodus 20:19). Thus when the leaders sinned, the entire nation was contaminated, and ultimately faced divine wrath. But the time is coming, Jeremiah says, when this proverb will be abandoned. "Instead," God promises, "everyone will die for his own sin; whoever eats sour grapes his own teeth will be set on edge" (Jeremiah 31:30). This could be true only if the entire covenantal structure associated with Moses‘ name is replaced by another. That is precisely what the Lord promises: he will "make a new covenant with the house of Israel and with the house of Judah that will not be like the covenant he made with their forefathers at the time of the Exodus." The nature of the promised new covenant is carefully recorded: God will put his law in the hearts and on the minds of his people. Instead of having a mediated knowledge of God, "they will all know me, from the least of them to the greatest," and therefore "no longer will a man teach his neighbor, or a man his brother, saying, 'Know the LORD‘' (31:31ff.). This does not foresee a time of no teachers; in the context, it foresees a time of no mediators, because the entire covenant community under this new covenant will have a personal knowledge of God, a knowledge characterized by the forgiveness of sin (31:34) and by the law of God written on the heart (31:33). . . .So I'd summarize it like this: in the old covenant, not everyone in the covenant community knew the Lord, and not everyone knew the Lord directly. In the new covenant, both change: everyone in the covenant community knows the Lord immediately and directly.
. . . the nature of the new covenant not be overlooked: as foreseen in the prophecy of Jeremiah, it is the abrogation of an essentially tribalistic covenantal structure in favor of one that focuses on the immediate knowledge of God by all people under the new covenant, a knowledge of God that turns on the forgiveness of sin and the transformation of the heart and mind.
Update: For those wanting to explore the credo understanding further, I'd recommend Steve Wellum's essay, Baptism and the Relationship between the Covenants (PDF). You can also read my interview with him.