What changed between the Old and New Testaments?

While I think that those who emphasise continuation are correct, there is nevertheless an unavoidable issue of supersession of the Law (or at least of an interpretation of it) from the Old Testament to the New Testament.

A constant theme of Christ's ministry is that it is not enough merely to follow the Law, or merely to act correctly simply by following precepts to the letter. It can be seen in a number of parables and sayings which privilege persons who do not follow the Law, but act with love and compassion, over those who do.

There is also a de facto Christian supersession of the Old Testament in that no Christian follows the majority of the laws laid out throughout it. Rather Christianity as a whole selects only the moral laws, typically ten, as a reminder.

This is not to say that God changed his mind, but rather that the old message was never truly one that emphasised the Law above all else, and that humanity in emphasising the Law and its ability to follow the Law scrupulously was acting out of pride.

Catholic and Scriptural Perspective

Answer

The Old is the type of the New, prefiguring it, with the New revealing and fulfilling the Old.

cf. The unity of the Old and New Testaments - Catechism of the Catholic Church

128 The Church, as early as apostolic times,¹ and then constantly in her Tradition, has illuminated the unity of the divine plan in the two Testaments through typology, which discerns in God's works of the Old Covenant prefigurations of what he accomplished in the fullness of time in the person of his incarnate Son.

129 Christians therefore read the Old Testament in the light of Christ crucified and risen. Such typological reading discloses the inexhaustible content of the Old Testament; but it must not make us forget that the Old Testament retains its own intrinsic value as Revelation reaffirmed by our Lord himself.² Besides, the New Testament has to be read in the light of the Old. Early Christian catechesis made constant use of the Old Testament.³ As an old saying put it, the New Testament lies hidden in the Old and the Old Testament is unveiled in the New.⁴

130 Typology indicates the dynamic movement toward the fulfillment of the divine plan when "God [will] be everything to everyone."⁵ Nor do the calling of the patriarchs and the exodus from Egypt, for example, lose their own value in God's plan, from the mere fact that they were intermediate stages.

And

IN BRIEF CCC 140 The unity of the two Testaments proceeds from the unity of God's plan and his Revelation. The Old Testament prepares for the New and the New Testament fulfills the Old; the two shed light on each other; both are true Word of God.

<sup>1. Cf. 1 Cor 10:6,11; Heb 10:l; l Pet 3:21.
2. Cf. Mk 12:29-31
3. Cf. 1 Cor 5:6-8; 10:1-11.
4. Cf. St. Augustine, Quaest. in Hept. 2,73:PL 34,623; Cf. DV 16.
5. 1 Cor 15:28.</sup>

Illustrative example from scripture

Heb 3:7-19 (RSVCE): How to reach God's land of rest

Warning against Unbelief

7 Therefore, as the Holy Spirit says,

“Today, when you hear his voice,
8 do not harden your hearts as in the rebellion, on the day of testing in the wilderness,
9 where your fathers put me to the test and saw my works for forty years.
10 Therefore I was provoked with that generation, and said, ‘They always go astray in their hearts; they have not known my ways.’
11 As I swore in my wrath, ‘They shall never enter my rest.’”^[a]

12 Take care, brethren, lest there be in any of you an evil, unbelieving heart, leading you to fall away from the living God. 13 But exhort one another every day, as long as it is called “today,” that none of you may be hardened by the deceitfulness of sin. 14 For we share in Christ, if only we hold our first confidence firm to the end, 15 while it is said,

“Today, when you hear his voice, do not harden your hearts as in the rebellion.”

16 Who were they that heard and yet were rebellious? Was it not all those who left Egypt under the leadership of Moses? 17 And with whom was he provoked forty years? Was it not with those who sinned, whose bodies fell in the wilderness? 18 And to whom did he swear that they should never enter his rest, but to those who were disobedient? 19 So we see that they were unable to enter because of unbelief.

Footnotes:

^{a. 3.11 Those who murmured against God in the desert were excluded from the promised land (the “rest”). Christians should beware lest, by offending God, they be excluded from heaven, the true rest, of which the promised land was a type.}

The footnote is a good summary example of the foregoing.

Is there any explanation for this in any doctrinal belief in any denomination which addresses this?

Jesus was explaining the limitation of the law. God's standard is holiness and perfection. Paul tells us that the law was supposed to bring us to an understanding that we needed a Savior. Instead many contemporaries of Jesus felt they had been successful in keeping the law.

What was needed was a new nature. Jesus admonishes Nicodemus for not knowing this.

This had been promised.

Jeremiah 31:33 But this is the covenant which I will make with the house of Israel: After those days, says the Lord, I will put My law within them, and on their hearts will I write it; and I will be their God, and they will be My people.