Both institutions had a glorious, truthful and divine beginning, but were at the degraded and corrupt end of a “dispensation” of the gospel (ie, Truth and Authority originally dispensed to men by God, but eventually corrupted by men) and at the cusp of a new. The leaders of both opposed the newer revelation/dispensation.
… The scribes and the Pharisees sit in Moses’ seat…
The Lord then put Peter in Moses’s seat.
By 1830, those who claimed to be rightful occupiers of Peter’s seat, along with those who claimed to be custodians of Truth while denying the need for a seat, were much like the scribes and pharisees of 30 A.D. (This is not to deny that there were some righteous and sincere men among those who claimed to have Authority or Truth.)
In 1830, the Lord put Joseph Smith Jr. in that seat. He was followed by Brigham Young.
Thomas S. Monson sits there now.