"But the ministry Jesus has received is as superior to theirs as the covenant of which he is mediator is superior to the old one, and it is founded on better promises. For if there had been nothing wrong with that first covenant, no place would have been sought for another." NIV translation
Just as the real thing is better than a picture or a copy, the new covenant in Jesus is superior to the Mosaic Covenant. In this passage, we see two reasons for its superiority. First, the new covenant has a far better promise in the fact that it guarantees us a restored relationship with God forever. We can know for certain that we will be with God for eternity. The second reason is that the first covenant required continual action on the part of man without man having the power to keep his part of the contract. The Law did not have the power to save as it could only show us our inability to keep it completely.
"But God found fault with the people and said: 'The time is coming, declares the Lord, when I will make a new covenant with the house of Israel and with the house of Judah. It will not be like the covenant I made with their forefathers when I took them by the hand to lead them out of Egypt, because they did not remain faithful to my covenant, and I turned away from them, declares the Lord.'" NIV translation
Paul quotes from the prophet Jeremiah where the people of Israel were told that a better covenant was coming. God explained that a new covenant was coming that did not require continual action on their part. He did this because they (and we) were unable to fulfill our part of the Mosaic Covenant.
"'This is the covenant I will make with the house of Israel after that time, declares the Lord. I will put my laws in their minds and write them on their hearts. I will be their God, and they will be my people. No longer will a man teach his neighbor, or a man his brother, saying, "Know the Lord," because they will all know me, from the least of them to the greatest. For I will forgive their wickedness and will remember their sins no more.''" NIV translation
God goes on to explain that this new covenant is superior because it promises that God will be with every believer through the Holy Spirit. Jesus referred to the Spirit as the Counselor and we see that it is the job of the Spirit to lead men in the things of God. He goes on to say how the Spirit will guide the believer and not the traditions of men or a religious ruling class.
Even today, there is a religious system that attempts to replace the work of the Spirit in the lives of believers. This is "the practices of the Nicolaitans" that Jesus said He hated (see Revelation 2:6).
"By calling this covenant 'new', he has made the first one obsolete; and what is obsolete and aging will soon disappear." NIV translation
This new covenant replaced the old which was the law. It is important to understand that it replaced the Law of Moses. In Paul's day, he was constantly facing those who wanted to hold onto the old covenant and blend it with the new. The same thing happens in the church of today and is the reason for the number of splits and factions (denominations) within the family of God. We must embrace the freedom that we have to live in Christ and let go of the old desire to do something to make ourselves more acceptable to God. When we don't embrace that freedom we, in fact, have fallen from grace.