What is the difference between the Levitical priesthood and the priesthood of Christ?

In his epistle to the Hebrews, St. Paul compares Aaron’s priesthood with Christ’s to show what the Church of the New Testament is enjoying as God’s kingdom on earth. The main points that the Apostle highlights are:

925 St. John Chrysostom, Homilies On Hebrews 11.3 (NPNF I/14:419).

Aaron and his descendants the priests became priests without an oath, but of Christ it was said, “The Lord has sworn and will not relent, you are a priest forever according to the order of Melchizedek by so much more Jesus has become a surety of better covenant” (Heb. 7:21– 22).

His capabilities: “[He] has come not according to the law of a fleshy commandment but according to the power of an endless life” (Heb. 7:16) The priests of the Old Testament had limited capabilities based on the law and descendancy. As for Christ, He is the only-begotten Son, whom death cannot overcome. Hence, the Church prays in the Sixth Hour of Good Friday saying, “Holy Mighty, who by weakness showed forth what is greater than power” (The Hymn of Omonogenees).

Christ’s Priesthood does not expire for He lives to fully save those who come forward through Him to the Father (Heb. 7:24). As for the Levite priests, they are limited by their lives that perish.

The Levitical priesthood offered many animal sacrifices (Heb. 9:12), but Christ offered His blood only once.

Through the law, a covenant was established with the house of Judah and the house of Israel (Heb. 8:7), where the believer discovers his shortcomings and need for the Savior who will elevate him to the heavens, “[He] raised us up together, and made us sit together in the heavenly places” (Eph. 2:6).

St. John Chrysostom says, “So also Christ did not glorify Himself to become High priest (Heb. 5:5). How then was He appointed (one says)? For Aaron was many times appointed as by the Rod (Ex. 7:8–13), and when the fire came down and destroyed those who wished to intrude into the priesthood. But in this instance, on the contrary, they [the Jewish Priests] not only suffered nothing, but they even rejoiced.”⁹²⁶