There is one apostle in Scripture that stands out as warranting our special attention...
How was the call of Saul like the call of the other eleven?
Saul did not receive his call from men. As the "twelve" were handpicked, so was Saul, through a direct message from Jesus Christ on the road to Damascus.
There is a question of witnesses, at first glance of the record. Who can testify that Saul had a miraculous encounter? The original 12 all had each other to confirm that, yes, Jesus had called them. We gather from the three accounts of his conversion in the Book of Acts that the men with him saw a great light, and heard something coming from the heavens, but did not necessarily distinguish the words being given Saul.
Thankfully, we are not dependent on these fellow-travelers, obviously men not friendly to the Gospel, for our needed witness. While God was working one miracle with Saul, another was taking place in the home of a disciple named Ananias. This brother received a vision from the Lord, telling him to go pray for the blinded Saul, and to be the vessel by which Saul would receive the Holy Spirit.
No, this thing was not done "in a corner." Many witnesses now come to the forefront of the story, seeing a man who has been totally transformed by the power of the risen Christ. His words, his actions, his life became the other witness in the story. Jesus literally shone from His new work in the one who had formerly been a Christ-hater.
Originally, as with the twelve, Saul was merely a disciple. He had so much to learn. He makes an interesting point of this in Galatians 1, saying that he did not get that training, even, from men. He went instead to the desert, where it seems so many of God's men must go. Paul was directly trained by the Lord!
Then he became a teacher, Acts 13:1. It was at that point, from the church at Antioch, that Saul, along with Barnabas as his companion, was sent out. The sending out and the equipping by the Holy Spirit seems to be the place where disciples and apostles are differentiated. So it was with the first followers of Jesus, and so with Saul.
How was the ministry of Saul/Paul different from others called "apostle" ?
By this time, Matthias had been sitting in the apostolic office. We know of no ministry he performed, though we doubt not that he was involved in ministry. Still, his calling was of men. We are told that his entry into the ministry was purely from the workings of Peter and the others. There was no supernatural intervention to announce his position, but only the casting of lots.
Barnabas, and later companions of Paul, were loosely called "apostles" also. Throughout the church's history there have been men who either gave themselves this name or were given it by a church that wanted to identify their ministry: The apostle to this or that country, the apostle with this or that mission.
But none of these men should probably be numbered among the twelve of Heaven. Jesus has, to this day, called only twelve men to serve as the foundation for his church.
Was Paul then the twelfth apostle? Did he fill Judas' office?
That is my opinion, based on the above similarities and differences noted. It is also interesting to observe again that throughout the Book of Acts there are never more than twelve men considered apostles. The book starts with eleven. Matthias brings the number to twelve for awhile. Then James moves on to Heaven. Notice that Peter makes no move to replace him as he had Judas. But it seems that God Himself brings the number back to twelve for the duration of the book by his special selection of the great apostle to the Gentiles.
Again I do not in any way suggest that this was common understanding in Jerusalem. Paul is asked to report there to the ten remaining apostles, and now James, the Lord's half-brother, in what seems to be a subservient manner. He continues to defend himself and his apostleship in Acts and in his epistles.
But it becomes clear, stepping back and looking at the panorama of the New Testament, that Paul has been added to the foundational group. Most of what has been passed on to us as foundational doctrine has come from this one man. His life measures up to that of the true apostle: chosen directly, scorned, authoritative, gifted, persecuted severely, martyred.
One other observation here: It seems to me that God could have used other men to do this work. But in separating Paul in this dramatic way, it seems He has lifted him up for us to see. From our vantage point it becomes clear who this man was intended to be. No such statement could be made of Matthias, Barnabas, Timothy, or any "apostle" since.