tl;dr> While it is true that Jesus never used the words "I am the Son of God"1 , many people around him did exactly as such. Thus, the difference between Jesus and Mansoor Hallaj is this - Hallaj claimed to be the Son of God on his own authority. Jesus was pro-claimed to be the Son of God by others.
1. Christians proclaim Jesus to be the Son of God as a matter of faith
Fundamentally, the difference between Jesus Christ and Mansoor Hallaj is that Christians believe Jesus of Nazerth to be the actual "only begotten Son" of God (John 3:16), and that as his son, he was the propitiation for our sins. As God, when He claimed to be "the Way, the Truth, and the Life" (John 14:6) and that "no one comes to the Father, but by me," we believe him to be accurate.
Fundamentally, this is a matter of faith. It is not subject to external scrutiny, for the Gospel is "foolishness to those who are perishing."
It is by definition not objective - for in the same way that if I told my wife "You are the most beautiful woman in the world," I could not "prove" that outside of my relationship with her. But, to those who are already, in faith, of the mind to believe that, the proof is self-evident.
It is simply a the claim, as expressed in Hebrews 1, that Jesus is in fact the Son, and is unique in that appelation. As Hebrews says of Jesus:
... in these last days [God] has spoken to us by his Son, whom he appointed heir of all things, and through whom also he made the universe. The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful word. After he had provided purification for sins, he sat down at the right hand of the Majesty in heaven. So he became as much superior to the angels as the name he has inherited is superior to theirs. For to which of the angels did God ever say, “You are my Son; today I have become your Father”? Or again, “I will be his Father, and he will be my Son”? And again, when God brings his firstborn into the world, he says, “Let all God’s angels worship him.”
2. Contemporary witnesses proclaimed Jesus to be the Son of God
That said, Christians make this claim, not because "someone decided they should" but rather based on the evidences they read in Scripture. For example, Peter famously confesses in Matthew 16:
"You are the Christ, the Son of the Living God."
Likewise, a Roman (i.e. - nono-Jewish non-"Christian") Centurion, upon witnessing the actual death of Jesus on the cross (and I know Muslims do not accept that he actually died before rising again), said almost the exact same thing:
When the centurion and those who were with him, keeping watch over Jesus, saw the earthquake and what took place, they were filled with awe and said, “Truly this was the Son of God!” (Matthew 27:54)
Even Jesus' enemies, the Pharisees, take account of these claims, and in effect, accuse Jesus of being God's son. Jesus, in reply says:
Why then do you accuse me of blasphemy because I said, 'I am God's Son'? Do not believe me unless I do what my Father does. But if I do it, even though you do not believe me, believe the miracles, that you may know and understand that the Father is in me, and I in the Father. (John 10:36-38)
3. External Authority is greater than a self-made claim
The difference between Jesus and Hallaj is thus a simple one: Hallaj claimed to be the Son of God, Jesus was claimed to be the Son of God by others.
It is similar to the story Jesus tells in Luke 5 - after Jesus had told a man his sins were forgiven, the Pharisees wondered under what authority he said this. The story continues:
...Jesus knew what they were thinking and asked, “Why are you thinking these things in your hearts? Which is easier: to say, ‘Your sins are forgiven,’ or to say, ‘Get up and walk’? But I want you to know that the Son of Man has authority on earth to forgive sins.” So he said to the paralyzed man, “I tell you, get up, take your mat and go home.” Immediately he stood up in front of them, took what he had been lying on and went home praising God. Everyone was amazed and gave praise to God. They were filled with awe and said, “We have seen remarkable things today.”
In short, while Hallaj may have suffered a death similar to Christ, and while he may have made claims similar to those of Christ, he was not in a position to do so. In Hebrews, we have God the Father (who is of one being with the Son), rhetorically saying of his Son, "Who else do I call my Son?" Rehetorically the answer is obvious - there is no else like Him. Others can say similar things, but ontologically He simply is the One who is God's Son, and he ontollogically is the Truth, because he God. In Matthew and Luke, we have Peter claiming that Jesus is God. And, in the Gospels, Jesus continually shows that he the authority of God, in the same manner a Son would have it. These form the basis on which Christians reject all other claims to being "the Son," "the Way," or "the Truth."
Any other claim is just words, not matter how compelling the narrative.
1 That said, Jesus did say "I and the Father are One," and when asked (Luke 22:70), "Are you the Son of God?" He replied, "You say that I am"