The big picture is that, if Adam failed to keep God's Law, when he's inside the Kingdom of God (living with God), it is expected that no humans can actually keep God's Law. This simply means all humans may have to die (second death) in front of God's Law ever since Adam. This is what Roman 5:14 trying to say. That is, even before Mosaic Law was put in place, death reigned over everyone including those who didn't break an obvious command as Adam did. It is because God's Law is always in place. And sin at anytime is a breaking of God's Law (the various forms of God's Law) one way or another.
At the same time, it says that God's Law is not completely equivalent to Mosaic Law. Mosaic Law is a covenantal Law. A covenant is composed of basically 3 important parts. 1) Law (covenantal law) and commandments, 2) Grace, and 3) a scope covering a certain group of humans. Mosaic Law is such a covenantal law attached to a covenant with a scope covering the Jews.
It is because humans ever since Adam cannot keep God's Law in complete (as witnessed and proven in the period between Adam and Noah), a covenant is needed for humans to be saved at all. A covenant simply means "You don't need to obey God's Law in complete, but you only need to obey the partial law specified in a covenant" such that you can be saved. Humans don't need to keep God's Law in full but only the covenantal law to be saved because God has grace for humans through each and every covenant. God doesn't need humans agreement but enforces a covenant to a group of humans simply because humans can only be saved through a covenant.
And the Grace part of a covenant is usually in the form of Faith, that is, God's Grace is usually granted through faith. And faith can save because in the end Jesus Christ will make a sacrifice. To simply put, humans don't need to (and can't) keep God's Law in full to be saved. They just need to keep what is said in a covenant. It is so because each covenant contains God's Grace through Jesus Christ.
As a result, the New (Second) Covenant is a far more advanced covenant. Whoever covered by this covenant no longer needs to subject himself to the judgment of Law. The Grace through Christ is complete in this covenant. This covenant is permanent and eternal. The Law/Commandment part of this covenant is kept to a minimal while its Grace is maximized that mostly you need only Faith in Christ to be saved.
The same verse also reflects the fact that after Adam being driven out of Eden, humans are no longer inside God's Kingdom and no longer living with God directly as Adam did. Under this circumstance, humans will be more and more sinful as time goes by, and generations after generations. In effect more and more humans will be "sentenced to death" in front of Law (covenantal Law). However, God will increase the "Grace" portion and decrease the "Law/Commandment" portion through each of His successive covenant, until the final one - the Second Covenant which is permanent and eternal! (with the Law/Commandment portion minimized and the Grace/Faith portion maximized).