What is the difference between Orthodox and Catholic understandings of apostolic succession?
There is indeed a different understanding, but it should be admitted that few are aware of it! For a good discussion, please see here:
Basically, in historic / patristic Orthodoxy, the bishop is Peter’s successor, while the presbyters stand for the apostles and do everything a bishop can do except ordain. The bishop is the symbol and center of unity of the Church (which is always the local Church).
In the Latin tradition (at least post-500 Latin tradition), all of the bishops are the successors to the Apostles and the Pope is Peter’s successor in this apostolic college. It is not clear whether the priests are part of this apostolic succession. This Latin model has been influential even in the Orthodox world, but the link above should be useful to understand the roots and “defenses” of both perspectives.