"What was wrong with the clothing they already had, and what was right about the clothing that God made for them?"
My short answer is "nothing" and "nothing". In fact, the question itself illustrates how differently mankind looks at life and how God looks at life.
I say that God wants to be with us. Genesis 3:8-9 talks about God seeking out Adam in the garden. Exodus 25:8 talks about God wanting to dwell with His people. John 3:16 talks about God's love for us. 1 Corinthians 1:9 talks about God calling us into fellowship.
When God was walking in the garden, Adam and Eve had already had their "tree" experience. God knew this. God still sought them out. God was not horrified by what they did. God did not turn His back on them. In fact, it was they that hid from God.
When asked by God, "where are you?", Adams reply was that he was naked and so he hid. For the first time shame was present. So was fear. Adam had declared that being naked was bad and wrong. Fear and shame followed this declaration.
When we declare something to be wrong, we want to separate ourselves from it. Adam did his best to separate himself from his nakedness by creating clothes.
However, God had no issue with their nakedness. Adam, even though he had already made a covering, still wanted to hide from God. God, wanting to be with Adam, made a more durable type of clothes for them so that their nakedness would not be an issue that would keep them apart.
Genesis 3:21 is where God makes clothes from animal skins. I say that neither Adam nor Eve would have thought to make clothes from the skin of an animal. Up to this point there was no talk of animals dying. Eating animals doesn't start until after the flood.
What we know is that after Adam, Eve and God talk, there is a dead animal and clothing. When Cain and Able have a fight about what sacrifice is acceptable to God, it is a dead animal and not the fruit of the ground that God accepts. This is because it was Adam and Eve who set up the agreement with God. God was sticking to this agreement, as God always does, when Cain offered part of his crop.
In keeping with how Adam and Eve kept shifting the blame in Gen. 3:12-13, they also blamed an animal. It was Adam and Eve who came up with the idea to sacrifice an animal to atone for their actions. God saw that they were keeping themselves from Him because of what they had done. God agreed to let Adam and Eve declare that an animal could atone for their actions and then God took that animal and made clothes.
God agreed to this so that He could be with them. Without it, they would continue to hide in fear. Doing something bad and wrong has always been an issue for mankind, not for God. God has always done His best to remove sin as a reason to keep away from Him.
God made clothes so that Adam and Eve wouldn't be present to shame and fear. God made clothes so that Adam and Eve could fellowship with God.