What in Deuteronomy implies that God gave those laws through Moses and not directly? According to the Catholic Church, what are the differences in the way the Mosaic/Deuteronomic Laws are given versus the Ten Commandments. I have been led to understand that there is a difference by some things I have heard, for instance the lack of a Theophany(?) in Deuteronomy - and that Jesus appears to upend many of the Deuteronomic standards in the New testament.

What are the differences between the Mosaic Law and the ten Commandments in terms of the circumstances of their being made binding?
If I'm understanding you, you are wondering about how some laws in the Old Testament differ, since Jesus seems to upend some, yet uphold others. And you also asked about how to know if they are binding. This answer hopefully will provide clarity to those points.

In the New Testament a distinction is made between commandments and the statutes (letter) of the Law (the Old Testament / Torah). Commandments are moral principles, personal and inter-personal. Statutes are practical ordinances dealing with possessions and actions.

The greek word for "commandments" was ἐντολῶν "entolon" (Concordance). And the word for "the Law" (the collection of laws in the Torah / Pentateuch) was τὸν νόμον "ton nomon" (Concordance). (In other places, νόμον alone could mean laws, rules, or principles in general, and not the entirety of the Pentateuch Law, depending on context). And finally, the greek words for individual laws are usually translated "statutes", "ordinances", or "decrees".

Here is an example that shows the conceptual difference between the commandments and the Law -- that not all statutes in the Law are commandments:


  "Circumcision is nothing and uncircumcision is nothing, but keeping God's commandments is what matters." (1 Corinthians 7:19)


In this sentence, you see that, while circumcision was an ordinance of the Law, it was not considered a commandment. So not all laws in the Law are commandments. 

Jesus also spoke of commandments, saying things like "whoever upholds these commandments and teaches others to do likewise will be called great in the kingdom of heaven" (Matt 5:19), and "you know the commandments -- you shall not murder, you shall not commit adultery.." (Mark 10:19)

The general principle is that God's commandments should be obeyed, whereas the statutes and laws in the Torah are not applicable anymore.

Here is a principle for how to determine the applicability of the Old Testament laws, and whether they are binding today:

All of the Pentateuch was given by God through Moses. But to whom? God specifically addresses the recipients in the beginning of many chapters -- the People of Israel. So none of the Pentateuch was given to non-Israelites.


  Then the LORD called to Moses and spoke to him from the tent of meeting, saying, "Speak to the sons of Israel and say to them, ...
  (Leviticus 1:1-2)


So in essence, God said, "Israelites, do these things." In saying this, he does not instruct anyone else besides the Israelites to do what he says.

But, God speaks truth among those statutes. He sometimes says "Israelites, do this, because this principle is true". Since God cannot lie, this means that "this principle" is true for all people. Even when he does not state his reasoning, we can meditate and wonder why he gave the laws as he did.

So since those laws were not spoken to us non-Israelites, then by default none of them are binding to us. Jesus bound the commandments to us, telling us to do them, but the rest of the Law remains not binding. But there is still much value and truth to be gained from reading and pondering the Old Testament laws.