How can that get more difficult? Well, never mind, let's list them explicitly. Mind, this is just about the nativity story.
1. Jesus' date of birth
According to Luke, during Quirinius' census, i.e., 6 CE.
According to Matthew, during Herod/s reign, i.e., 4 BCE or earlier.
2. Residence of Joseph and Mary (before Jesus' birth)
According to Luke, Nazareth in the Galilee
According to Matthew, Bethlehem in Judea
3. Whereabouts of the Holy Family after Jesus' birth
According to Luke, they present Jesus in the Jerusalem temple and return to Nazareth.
According to Matthew, they flee to Egypt, then return to Israel and settle in Nazareth for fear of the Judean ethnarch, Archelaus.
4. Star & Magi
Matthew mentions the star and the magi.
Luke, the eminent historian of natural phenomena according to Sir Willam Ramsay, doesn't mention them.
5. Massacre of the Innocents
Matthew mentions this.
Luke, the eminent historian (etc.), doesn't mention it.
And let's throw in a bonus for good measure:
6. Genealogy of Jesus
Matthew 1:1-17 has a different genealogy than Luke 3:23-37. Hell, they even differ in who is Joseph's father.