When I research on the Catholic Priesthood, many apologists and theologians will make comparisons between Israelite priests and Catholic priests, especially in reference to the Pope guiding the church.
What are the significant differences in role and authority between the Israelite priesthood and the Catholic priesthood?
St. Thomas Aquinas explains the difference between these two priesthoods when he answers "Whether the priesthood of Christ was according to the order of Melchisedech?" (Summa Theologica III q. 22 a. 6 co.):
the priesthood of the [Old] Law was a figure of the priesthood of Christ, not as adequately representing the reality, but as falling far short thereof: both because the priesthood of the Law did not wash away sins, and because it was not eternal, as the priesthood of Christ. Now the excellence of Christ's over the Levitical priesthood was foreshadowed in the priesthood of Melchisedech, who received tithes from Abraham, in whose loins the priesthood of the Law was tithed. Consequently the priesthood of Christ is said to be "according to the order of Melchisedech" (Ps. 109:4), on account of the excellence of the true priesthood over the figural priesthood of the Law.
Differences between Levitical and Melchizedek priesthoods:
Qualification:
Bloodline Levitical
Whosoever Melchizedek
Duration:
Temporary Levitical
Permanent Melchizedek
Efficacy of sacrifice:
Year to year Levitical
Eternal Melchizedek
Calling:
Earthly Levitical
Heavenly Melchizedek
Sabbath rest
Weekly Levitical
Always Melchizedek
Made by God's Oath
No Levitical
Yes Melchizedek
Needing to offer daily sacrifices
Yes Levitical
No Melchizedek
Minister of the true tabernacle
No Levitical
Yes Melchizedek
Faultless covenant
No Levitical
Yes Melchizedek
Blood sacrifice
Animals/other Levitical
Jesus Christ Melchizedek
Source of blood
Others/animals Levitical
Self Melchizedek
Number of sacrifices required
Thousands Levitical
One Melchizedek
