I was brought up Catholic/Presbyterian in Scotland and my understanding from Sunday School, sermons, Gospel readings etc. is:
It’s not an inter-testament change. The change happens in the New Testament, when Jesus who is the Son of God rewrites both the Law and the Covenant.
For the Law, he distils all the commandments down to two: Honour God, and love thy neighbour. This is attributed to him directly in the Gospels, and Paul goes very big on it in his many letters. Jesus also says that in him the Laws are fulfilled and although Jesus does also say in Matthew that the Laws will be with us until the end of time, Paul again takes the first idea and runs with it, saying that once Jesus fulfilled his purpose through crucifixion and resurrection, he abolished the old laws, leaving only the two above.
The second part is the Covenant. The Christian Old Testament God made a deal with the Jews, to the effect that in order to be righteous they had to obey His Laws. Jesus offered a new deal. Through his sacrifice, akin to but infinitely more sacred than sacrificial offerings of animals, the slate was wiped clean and now the only way to be in a state of grace was to follow the teachings of/believe in Jesus. “I am the Way, the Truth, and the Life, none come to the Father except through me.”
The other way that Jesus rewrites the Covenant is that he places both reward and punishment for following God’s teachings firmly in the afterlife. If you are meek and peaceloving and give away all your possessions and turn the other cheek like he says you should do, this will not necessarily lead to material reward. But when you get to Heaven, and the sheep are separated from the goats, you will get your reward from the many mansions there.
So Jesus enacts/represents a major rewriting of humanity’s relationship with the divine.
Therefore Christians reading the Old Testament see a “Father” who both has continuity with the God they worship but also has a very different relationship with His people. It’s a relationship based on strict rules, for transgressions of which strict and worldly punishments (pillar of salt, fire and brimstone, eaten by bears etc.) are meted out without very much in the way of mercy. The benefits of following the rules are also material - the land of milk and honey, the laying low of enemies, the protection and prosperity of the Jewish people are all on offer in this world.
The distinctive and to some extent contrasting natures of the New and Old Testament divinities is addressed by the notion of the Trinity - God the Father (very much a Patriarch who demands sacrifice from you) and God the Son, who offers Himself in sacrifice for you. By no means is God the Father over and done with, but the relationship between humanity and God has changed.