Alma the Younger and the sons of Mosiah were going around corrupting people, and turning them from the teachings of God. An angel appeared to them and they completely changed. They became great teachers for Christ. Paul (Saul at the time) went about imprisoning members of the church. Christ appeared to him, and he changed to eventually become a very powerful missionary. Many saw Christ's miracles and therefore listened to his words.
So why doesn't God just convert everyone like this?
Let's look at the other side of this for a moment. God showed great miracles in Egypt at the hands of Moses. Yet we get the impression that no Egyptians joined Israel. Even the Israelites did nothing but winge. Even when they got out of Egypt they continued to disbelieve and winge. They saw the miracle of water coming after Moses struck a rock. I'm sure you know I could go on and on with the miracles they saw. Did those who saw what happened at the Tower of Babel suddenly repent? And what of Daniel in the lion's den. Did the Babylonians take up following God because of this? The scribes and pharasees saw the miracles of Christ also. Did they believe in him and repent? In spite of all the miracles Christ demonstrated, when he told his followers who he really was many left (John 6:66).
In Luke 16:19-31 Christ gives the parable of Lazareth and the rich man. While there are some interesting concepts in it, the point of the parable was stated in the last verse. "And he said unto him, if they hear not Moses and the prophets, neither will they be persuaded, though one rose from the dead." This refutes the idea that if God appeared to someone they would believe.
So how does this fit with Paul and Alma etc? They genuinely believed they were doing the right thing. They went out with fervent effort to right things with what must have been open minds, just not catching the point. So these appearances opened them up to hear. But people are not normally like this. People convince themselves that they are genuine, but true examination would reveal differently, or God would have them see a miracle.