Each of the gospels shows Jesus Christ from a different vantage point. The four different portraits of Jesus Christ are taken from Ezekiel 1:10 (As for the form of their faces, each had the face of a man; all four had the face of a lion on the right and the face of a bull on the left, and all four had the face of an eagle.) The Gospel of Matthew shows Jesus to be King or face of the lion; Mark shows Jesus to be Man or the face of a man; Luke shows Jesus to be the Servant or the face of the bull; and John shows Jesus to be God or the face of the eagle. Each author had a purpose and an audience when writing their portrayal of Jesus. As stated in the question, Matthew to the Jews, Mark to the Romans, Luke to the Greeks, and John to the world. The Synoptic Gospels (Matthew, Mark, and Luke) differ from the Gospel of John; Matthew and Mark share 90 percent of Mark’s material but only 8 percent of John’s material is consent with the other three gospels. John wrote his gospel in Ephesus in Asia with a target audience of diaspora Jews and Gentiles attracted to Judaism. The differences between the Synoptic Gospels and John can be due to John’s purpose for writing his gospel as stated in John 20:30-31;
“Therefore many other signs Jesus also performed in the presence of the disciples, which are not written in this book; but these have been written so that you may believe that Jesus is the Christ, the Son of God; and that believing you may have life in His name.”
Through the “signs” that Jesus did point to the fact the He is the Christ or promised One. As noted in John 20:31, John’s twofold purpose for writing was to show Jesus through His miracles (or signs) and teachings so people might, first, believe that He was indeed who He said He was, the Son of God; and second, have eternal life because of their belief. John wrote to show Jesus Christ to be the Jewish Messiah that was prophesized to come in the Old Testament.
Since we know the audience of John to be diaspora Jews and Gentiles attracted to Judaism, the only reason a Jewish scholar would make a statement like, “To us Jews, the Fourth Gospel is the most Jewish of the four!” would be because John showed that Jesus was the Messiah; the promised One. The signs John showed were met to elicit faith in Jesus as Messiah. The purpose for these signs was to show Jesus’ power over creation, space, time, food, natural laws, physical laws, death, and over all the above. Being a Jew, John uses what he knows and what he saw to show Jesus to be Messiah. He quotes occasionally from the Hebrew text (cf. 12:40; 13:18; 19:37); he was acquainted with the Jewish feasts such as the Passover (2:13; [5:1]; 6:4; 11:55), Tabernacles (7:37), and Dedication/Hanukkah (10:22); he was acquainted with Jewish customs such as the arranging of water pots (ch. 2) and burial customs (11:38-44).
Many people tell new converts to begin by reading John because of its profound simplicity. The writings of John may be compared to a well of water, so clear and sparkling that at first one thinks he sees to the bottom; but that well is so deep that the more a person gazes into it, the deeper it appears. No one has yet to be able to fathom it.
 Andreas J. Kostenberger, Encountering John. (Grand Rapids, MI: Baker Academic, 1999), 198.
 Ibid, 25.
 Leon Morris, Jesus is the Christ: Studies in the Theology of John. (Grand Rapids, MI: William B. Eerdmans Publishing Co., 1989), 1.
 Elmer Towns, The Gospel of John: Believe and Live. (Chattanooga, TN: AMG Publishers, 2002), xi.
 Leon Morris, Jesus is the Christ: Studies in the Theology of John. (Grand Rapids, MI: William B. Eerdmans Publishing Co., 1989), 4.
 Elmer Towns, The Gospel of John: Believe and Live. (Chattanooga, TN: AMG Publishers, 2002), xiii.
 Daniel B. Wallace, “The Gospel of John: Introduction, Argument, Outline,” Bible.org, June 28, 2004. https://bible.org/seriespage/gospel-john-introduction-argument-outline
 Elmer Towns, “Introduction to John: Christ – The Word of God” (presentation, Module One of BIBL 323 John Liberty University, 2014).