Why are there differences in the Gospels? Are they historically reliable?

First of all, the gospels we have in the New Testament are not eyewitness accounts. They are all anonymous and we don't know who wrote them. The title of the gospel was added in the second century CE to make distinctions among the different gospels. We believe that they were written down at least 40-60 years after Jesus died. All of them were written in Greek. This is odd given the fact that Jesus and his disciples spoke Aramaic. Jesus's disciples were Galileans and illiterate; therefore, they could not write in Greek. Until the gospels were written down by Greek-speaking persons, a variety of oral traditions about Jesus had been circulating in different regions beyond Palestine, including Greek-speaking regions.

So depending on who wrote or in which community the Evangelist worked, the content of the gospel changed. Therefore, changes in the gospels are expected. Sometimes, the Evangelists (Mark, Matthew, Luke, and John) tell the same story about Jesus differently. For example, Jesus's water baptism in the synoptic gospels is told differently yet similarly. Jesus's crucifixion between Mark and Matthew is told very differently. In Mark, Jesus worries about his death, being shocked and agitated by his impending death; he cries at the last minute: Eli Eli lama Sabachthani, which means, "my God, my God, why have you forsaken me?" This portrait of Jesus reflects Markan theology about the suffering of Jesus. But in Luke, he is so calm and adamant about his death because he knew how he was going to die as a prophet. He prays for those who kill him, saying "Father, forgive them because they don't know what they do." He dies as a great hero of the prophet, who says: "Into your hands, I commit my spirit." Many events or things in the gospels are told differently though the degree is different. Again, this is because the Evangelists edited the inherited sources and registered their theological voices/interpretations that reflect their communities.

But there are also irreconcilable accounts between the Gospels. In this case, one account may be right or both of them may be incorrect. Otherwise, not both of them can be correct.
1) Where is Jesus's hometown? (Nazareth in Luke; Bethlehem in Matthew);
2) Which chronology of Jesus is correct? (In Matthew: Jesus's parents flee to Egypt from Bethlehem because of Herod's threat; In Luke, they return to Nazareth after Jesus's birth in Bethlehem);
3) Where do Jesus's disciples have to gather after the resurrection? (In Mark/Matthew: Galilee; but in Luke, they should stay in Jerusalem);
4) When does Jesus die? In John, he was crucified on the day of preparation for Passover (Thursday); in Mark, he was crucified after the Passover meal was eaten, which is the last supper (Friday).
5) There are also a lot of differences in Jesus's resurrection account.

Then, are the gospels historically reliable? If you study the historical Jesus, you should treat the gospels carefully. Some materials will be helpful, while others are not so. There are some criteria by which we can tell which texts may preserve the closeness of Jesus. This is the area of study we call the historical Jesus.

From the religious perspective, we need to study each gospel on its own, trying to understand the Evangelist's community and theological agenda. For example, we may think of four different images of Jesuses in the four gospels. The Markan Jesus "came not to be served, but to serve and give his life" (Mk 10:45). The Matthean Jesus came not to destroy the law and prophets but to fulfill (Matt 5:17). The Lukan Jesus came to seek out and save the lost (Lk 19:10). The Johannine Jesus was born as a king to testify to the truth of God (Jn 18:37).

The bottom line is there is an unbridgeable gap between the gospels and the historical Jesus. Nevertheless, there are some clues by which we may understand him better than otherwise. Essentially, our job is to seek meaning both in Jesus's time and in the later gospel communities' time.

Among my works, the following books deal with the above issues: (1) Resurrecting Jesus: The Renewal of New Testament Theology; (2) Messiah in Weakness: A Portrait of Jesus from the Perspective of the Dispossessed.