The Gospel of Luke begins its presentation of Jesus' ministry with a kind of "inaugural address" Jesus gives in his home synagogue of Nazareth. He picks up the scroll of Isaiah and reads from chapter 61: "The Spirit of the Lord is on me, because he has anointed me to proclaim good news to the poor. He has sent me to proclaim freedom for the prisoners and recovery of sight for the blind, to set the oppressed free, to proclaim the year of the Lord’s favor" (Luke 4:18-19). If each gospel has its own special emphases, this is one on which Luke especially focuses.
On an occasion early in Mark, Jesus is criticized for hanging out with "sinners." His response is that "It is not the healthy who need a doctor, but the sick. I have not come to call the righteous but sinners" (Mark 2:17). Jesus has gone to the house of a tax collector named Levi. Tax collectors were known not only for taking much needed resources from the common person for the powers they worked for. They were known for getting rich themselves by adding on to the amount taken. Jesus never disputes that they are sinners. What he disputes is the sense that they were unredeemable.
Most of us know the Parable of the Prodigal Son, but Matthew 21 gives us a shorter version that captures the bottom line well: "There was a man who had two sons. He went to the first and said, 'Son, go and work today in the vineyard.' 'I will not,' he answered, but later he changed his mind and went. Then the father went to the other son and said the same thing. He answered, 'I will, sir,' but he did not go. Which of the two did what his father wanted? 'The first,' they answered" (Matt. 21:28-29).
The two sons correspond to the "healthy" and the "sick" respectively. The sick are the tax collectors and prostitutes Jesus mentions immediately after telling this short parable (Matt. 21:31-32). The "healthy," by contrast, are those who have just asked Jesus who gave him the authority to do the things he did. In this context, the chief priests and elders, the leaders of Jerusalem, are talking to him (21:23). But at other times in the gospels, he refers to Pharisees and other teachers as the healthy (e.g., Mark 2:16).
Our heads are so full of things from Christian tradition and other parts of the New Testament (e.g., Paul) that it is hard for us to hear the connotations words like "healthy" and "sinner" likely had at the time of Jesus...