And now he hath obtained a more excellent ministry — His priesthood as much excels theirs, as the promises of the gospel (whereof he is a surety) excels those of the law. These better promises are specified, Hebrews 8:10-11: those in the law were mostly temporal promises.
Related Commentaries of Hebrews 8:6
Hebrews 8:6
6 But now hath he obtained a more excellent ministry, by how much also he is the mediator of a better covenant,b which was established upon better promises.