Bs"d
No, there is no difference. What you have to keep in mind is that there is no "law of Moses", what there is are many laws of God, given through Moses. Therefore it is called "law of Moses", but it is the law of God.
Here are some examples: "Now Y-H-W-H spoke to Moses and Aaron, saying to them, 2 “Speak to the children of Israel, saying, ‘These are the animals which you may eat among all the animals that are on the earth: 3 Among the animals, whatever divides the hoof, having cloven hooves and chewing the cud—that you may eat. 4 Nevertheless these you shall not eat among those that chew the cud or those that have cloven hooves: the camel, because it chews the cud but does not have cloven hooves, is unclean to you; 5 the rock hyrax, because it chews the cud but does not have cloven hooves, is unclean to you; 6 the hare, because it chews the cud but does not have cloven hooves, is unclean to you; 7 and the swine, though it divides the hoof, having cloven hooves, yet does not chew the cud, is unclean to you. 8 Their flesh you shall not eat, and their carcasses you shall not touch. They are unclean to you."
Lev 11
So you see that the law about not eating pork is given by God, but given to the Israelites by Moses and Aaron, therefore it is called the law of Moses, but it is the law of God.
Another one:
"Then Y-H-W-H spoke to Moses, saying, 2 “Speak to the sons of Israel, saying:
‘When a woman gives birth and bears a male child, then she shall be unclean for seven days, as in the days of her menstruation she shall be unclean. 3 On the eighth day the flesh of his foreskin shall be circumcised."
Lev 12
There are hundreds more examples like this in the Torah, the five books of Moses.
The 10 commandments were given straight by God to the Israelites. Because the Israelites were very afraid, the rest came through Moses.
In the NT no difference is made between the 10 commandments and the law of Moses.
When the Christian messiah speaks about "the law" he means the five books of Moses, containing all 613 commandments.
The Jewish division of the Tanach, the Hebrew Bible, is in three parts: The Torah, (T) the prophets, (in Hebrew: nevi'iem, the N) and the writings, (in Hebrew Chetuviem, the Ch, making for the acronym "Tanach")
The Torah are the five books of Moses. The prophets are the prophetic books, the big and small prophets, and the writings are books like Ruth, Esther, Proverbs, etc, and that section begins with the Psalms.
And we see that in the NT that exact same division is made, when it is said: "Then He said to them, “These are the words which I spoke to you while I was still with you, that all things must be fulfilled which were written in the Law of Moses and the Prophets and the Psalms concerning Me.”
Luke 24:44
So here "the law of Moses" means the first five books of the Bible, Genesis, Exodus, Leviticus, Numbers, and Deuteronomy, which include the 10 commandments.
When the question is asked, "What is the first commandment?", not a commandment from the 10 commandments is given, but from the law of Moses:
"Then one of the scribes came, and having heard them reasoning together, perceiving that He had answered them well, asked Him, “Which is the first commandment of all?”
29 Jesus answered him, “The first of all the commandments is: ‘Hear, O Israel, the Lord our God, the Lord is one. 30 And you shall love the Lord your God with all your heart, with all your soul, with all your mind, and with all your strength.’ This is the first commandment. 31 And the second, like it, is this: ‘You shall love your neighbor as yourself.’ There is no other commandment greater than these.”
Mark 12
"Hear, O Israel, the Lord our God, the Lord is one." is Deut 6:4, and "You shall love your neighbor as yourself" is Lev 19:18, both from the law of Moses, not from the 10 commandments.
So no difference is made.
"Serve Y-H-W-H! And if it seems evil to you to serve Y-H-W-H, choose for yourselves this day whom you will serve, whether the gods which your fathers served that were on the other side of the River, or the gods of the Amorites, in whose land you dwell.
But as for me and my house, we will serve Y-H-W-H!”
Joshua 24:14-15