The Ten Commandments were part of the
Kingdom of Priests, a Holy nation and God's Possession among the nations Covenant and is first told to the Israelite nation in Ex 20. The Ex 34 reference you are referring too, is where the Ten Commandments that form part of the
Kingdom of Priests, a Holy nation and God's Possession among the nations Covenant is recorded by God on tablets of Stone and given to Moses.
What you are really asking, based on your response above is, "
What is the difference between the Old and the Refurbished Ten Commandments by Christ?"
The answer is the Old form consisted of ten commandment statutes while the refurbished commandments of Christ consisted of two statutes on which hang all of the law and the prophets.
But Christ also said: -
Matt 5:17-19: - 17 "Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill. 18 For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled. 19 Whoever therefore breaks one of the least of these commandments, and teaches men so, shall be called least in the kingdom of heaven; but whoever does and teaches them, he shall be called great in the kingdom of heaven.
Luke 16:16-17: - 16 "The law and the prophets were until John. Since that time the kingdom of God has been preached, and everyone is pressing into it.
17 And it is easier for heaven and earth to pass away than for one tittle of the law to fail.
Now, my question is: -
What covenant did Christ come and preach about?
Shalom