Matthew's Gospel carries with it a greater respect for what is seen, respecting the Jewish mentality of the audience that he was writing to, which is under law.
First, there was the Centurion:
"5And when Jesus was entered into Capernaum, there came unto him a centurion, beseeching him, 6And saying, Lord, my servant lieth at home sick of the palsy, grievously tormented. 7And Jesus saith unto him, I will come and heal him. 8The centurion answered and said, Lord, I am not worthy that thou shouldest come under my roof: but speak the word only, and my servant shall be healed. 9For I am a man under authority, having soldiers under me: and I say to this man, Go, and he goeth; and to another, Come, and he cometh; and to my servant, Do this, and he doeth it. 10When Jesus heard it, he marvelled, and said to them that followed, Verily I say unto you, I have not found so great faith, no, not in Israel. 11And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven. 12But the children of the kingdom shall be cast out into outer darkness: there shall be weeping and gnashing of teeth. 13And Jesus said unto the centurion, Go thy way; and as thou hast believed, so be it done unto thee. And his servant was healed in the selfsame hour." (Matthew 8: 5-13)
Here is the account in Luke:
"2And a certain centurion's servant, who was dear unto him, was sick, and ready to die. 3And when he heard of Jesus, he sent unto him the elders of the Jews, beseeching him that he would come and heal his servant. 4And when they came to Jesus, they besought him instantly, saying, That he was worthy for whom he should do this: 5For he loveth our nation, and he hath built us a synagogue. 6Then Jesus went with them. And when he was now not far from the house, the centurion sent friends to him, saying unto him, Lord, trouble not thyself: for I am not worthy that thou shouldest enter under my roof: 7Wherefore neither thought I myself worthy to come unto thee: but say in a word, and my servant shall be healed. 8For I also am a man set under authority, having under me soldiers, and I say unto one, Go, and he goeth; and to another, Come, and he cometh; and to my servant, Do this, and he doeth it. 9When Jesus heard these things, he marvelled at him, and turned him about, and said unto the people that followed him, I say unto you, I have not found so great faith, no, not in Israel. 10And they that were sent, returning to the house, found the servant whole that had been sick." (Luke 7: 2-10)
Now, these two passages may seem to conflict with each other. Matthew's account speaks of the centurion actually coming and speaking to Jesus, while Luke's account writes about "the centurion's servant." What's going on?
Simply put, Matthew was an imperial tax collector, and as such he recognized that the centurion's servant was coming to Jesus bearing the authority of the centurion, most likely the ring. Luke, on the other hand, wrote down exactly what he saw: the centurion's servant.
Other discrepancies are worthy exploring. Matthew's Gospel includes the following account:
11And I say unto you, That many shall come from the east and west, and shall sit down with Abraham, and Isaac, and Jacob, in the kingdom of heaven. 12But the children of the kingdom shall be cast out into outer darkness: there shall be weeping and gnashing of teeth.
I presume that Matthew included this part in order to provoke his Jewish audience to jealousy:
"But I say, Did not Israel know? First Moses saith, I will provoke you to
jealousy by them that are no people, and by a foolish nation I
will anger you." (Romans 10: 19)
Luke's Gospel includes a unique element:
"And when he heard of Jesus, he sent unto him the elders of the Jews, beseeching him that he would come and heal his servant. 4And when they came to Jesus, they besought him instantly, saying, That he was worthy for whom he should do this: 5For he loveth our nation, and he hath built us a synagogue. "
I believe that Luke included this element as an indication of the risk that the Centurion took to reach out to Jesus, so unwilling to offend Him, having such respect for Jesus the Messiah, a Rabbi while on earth who still operated according to the law of Moses, according to the Centurion. I also believe that Luke wanted to emphasize the differences between Jew and Gentile based on the law, providing a fuller account for his readers.
Another verse which is missing in Luke's account yet provided for us in Matthew's:
13And Jesus said unto the centurion, Go thy way; and as thou hast believed, so be it done unto thee. And his servant was healed in the selfsame hour.
I believe that Matthew included this verse, prompted by the Holy Spirit, in order to impress on his Jewish readers that Jesus is looking for faith, not works.