"The former regulation is set aside because it was weak and useless (for the law made nothing perfect), and a better hope is introduced, by which we draw near to God." NIV translation
The law establishing the Levitical priesthood did not give the priests the power to defeat sin. Instead, they were constantly offering sacrifices and in fact became slaves to the sacrifices and the law. But, a better hope (speaking of Jesus Christ) was introduced as He overcame sin and death on our behalf. Through Him, we can draw near to God.
"And it was not without an oath! Others became priests without any oath, but he became a priest with an oath when God said to him: 'The Lord has sworn and will not change his mind: "You are a priest forever."' Because of this oath, Jesus has become the guarantee of a better covenant." NIV translation
In Psalm 110, God used David to describe Jesus as the coming King. Unlike the world where there is separation between government and the reign of God, He was also described as "a priest forever in the order of Melchizedek". This promise of God is the guarantee of the new covenant of grace that replaced the Law.
"Now there have been many of those priests, since death prevented them from continuing in office; but because Jesus lives forever, he has a permanent priesthood. Therefore he is able to save completely those who come to God through him, because he always lives to intercede for them." NIV translation
One of the greatest differences between Jesus and the Levitical high priests is that they died but He lives forever. They could only serve for a short time but He is still serving as He is at the Father's right hand.
"Such a high priest meets our need - one who is holy, blameless, pure, set apart from sinners, exalted above the heavens. Unlike the other high priests, he does not need to offer sacrifices day after day, first for his own sins, and then for the sins of the people. He sacrificed for their sins once for all when he offered himself. For the law appoints as high priests men who are weak; but the oath, which came after the law, appointed the Son, who has been made perfect forever." NIV translation
The Levitical priests had to make atonement for their own sins before they could minister on behalf of the people. Since they continued to sin, this was an unending process. Jesus who did not sin did not require atonement but instead became the perfect sacrifice for all men.