The centurion's servant
Whether the centurion came to Jesus in person, as Matthew has it, or sent elders, as Luke does.
Matthew 8:5-13
5When he came into Capernaum, a centurion came to him, asking him,
6and saying, “Lord, my servant lies in the house paralyzed, grievously tormented.”
7Jesus said to him, “I will come and heal him.”
8The centurion answered, “Lord, I’m not worthy for you to come under my roof. Just say the word, and my servant will be healed.
9For I am also a man under authority, having under myself soldiers. I tell this one, ‘Go,’ and he goes; and tell another, ‘Come,’ and he comes; and tell my servant, ‘Do this,’ and he does it.”
10When Jesus heard it, he marveled, and said to those who followed, “Most certainly I tell you, I haven’t found so great a faith, not even in Israel.
11I tell you that many will come from the east and the west, and will sit down with Abraham, Isaac, and Jacob in the Kingdom of Heaven,
12but the children of the Kingdom will be thrown out into the outer darkness. There will be weeping and gnashing of teeth.”
13Jesus said to the centurion, “Go your way. Let it be done for you as you have believed.” His servant was healed in that hour.
Luke 7:1-10
1After he had finished speaking in the hearing of the people, he entered into Capernaum.
2A certain centurion’s servant, who was dear to him, was sick and at the point of death.
3When he heard about Jesus, he sent to him elders of the Jews, asking him to come and save his servant.
4When they came to Jesus, they begged him earnestly, saying, “He is worthy for you to do this for him,
5for he loves our nation, and he built our synagogue for us.”
6Jesus went with them. When he was now not far from the house, the centurion sent friends to him, saying to him, “Lord, don’t trouble yourself, for I am not worthy for you to come under my roof.
7Therefore I didn’t even think myself worthy to come to you; but say the word, and my servant will be healed.
8For I also am a man placed under authority, having under myself soldiers. I tell this one, ‘Go!’ and he goes; and to another, ‘Come!’ and he comes; and to my servant, ‘Do this,’ and he does it.”
9When Jesus heard these things, he marveled at him, and turned and said to the multitude who followed him, “I tell you, I have not found such great faith, no, not in Israel.”
10Those who were sent, returning to the house, found that the servant who had been sick was well.
What each tradition says
Rabbinic / Midrashic
No sourced position found. Checked the Jewish Encyclopedia (articles 'Jesus', 'Gospel', 'New Testament': no treatment of the centurion of Capharnaum), Sefaria's Midrash Rabbah, Targum Jonathan and Rashi (no Gospels material) — no rabbinic statement on Matt. viii. 5-13 / Luke vii. 1-10 found.
Patristic
Augustine acknowledges an apparent discrepancy: if the centurion did not come in person but sent friends, how can Matthew say he came to Jesus? Augustine holds that the discrepancy disappears on careful examination, because Matthew simply used a familiar mode of expression.
Augustine of Hippo (354-430), On the Harmony of the Gospels, Book II (trans. S. D. F. Salmond), in Nicene and Post-Nicene Fathers, ser. 1, vol. 6 (New York: Christian Literature Publishing Co., 1888) — Book II, ch. 20, §49 ('An Explanation of the Circumstance that Matthew Tells Us How the Centurion Came to Jesus...')
Catholic
George J. Reid states that over against the contention sometimes maintained with great erudition, the Church of God has always proclaimed belief in the historical accuracy and consequent real harmony of the canonical Gospels. Her doctors, notably Eusebius, Jerome, and Augustine, and commentators have invariably professed that belief.
George J. Reid, The Catholic Encyclopedia (New York: Robert Appleton Company), vol. 6, s.v. 'Gospel and Gospels' — article 'Gospel and Gospels', section 'Divergences of the gospels'
Orthodox
Chrysostom observes that the leper came to Jesus when He had come down from the mountain, but the centurion when He entered Capernaum. He explains that neither went up the mountain, not from remissness—since both had fervent faith—but so as not to interrupt His teaching.
John Chrysostom (c. 347-407), Homilies on the Gospel of St. Matthew (trans. George Prevost), in Nicene and Post-Nicene Fathers, ser. 1, vol. 10 (New York: Christian Literature Publishing Co., 1888) — Homily XXVI (on Matthew 8:5-13), opening paragraph
Reformed / Protestant
Calvin notes that the only difference is Matthew saying the centurion came to him, while Luke says he sent some of the Jews to plead in his name. Calvin sees no impropriety in Matthew saying the centurion did what was done in his name and at his request.
John Calvin (1509-1564), Commentary on a Harmony of the Evangelists, Matthew, Mark, and Luke, vol. 1 (trans. William Pringle; Edinburgh: Calvin Translation Society, 1845) — Commentary on Matthew 8:5 ('And when Jesus had entered')
Evangelical
Adam Clarke describes the centurion as a Roman military officer who had the command of one hundred men.
Adam Clarke (c. 1760-1832), The Holy Bible... with a Commentary and Critical Notes (1810-1826; 'Adam Clarke Bible Commentary') — Commentary on Matthew 8:5 ('A centurion')