In the first case it looks like the daughter has just perished, in the second, she's about to. How to resolve this?
The answer is found in another question. Do you believe that every word Jesus said is true and fully trustworthy? I'm going to assume a "Yes" because otherwise I'm not dealing with someone who knows Jesus can't lie (Heb 6:18) and that calls for an entirely different blog post, attempting to convince someone Jesus is who He says and not only can be trusted, but must be trusted.
So yes, He is true in all His ways. We then come to John 10:35 and read, "Scripture cannot be broken." Now since we have God Himself saying that Scripture is perfect, and cannot be anything other than truth it stands to reason it cannot contradict itself, it cannot be a lie, and it cannot be proven false. It's literally breathed out by God, useful for teaching, rebuking, training in righteousness.
And if that is believed then the different accounts of Jairus daughter can easily be understood, because it doesn't take much effort to reconcile the record if you already believe that they were complementary rather than in opposition.
But you still ask, why are they different? The answer is that the gospel accounts are trying to emphasize different things. Matthew is written to the Jews, it's always using the word 'king' - there was a certain king, the kingdom of heaven is like, etc. It's thrust is to teach the Jews the right doctrine and lessons to show them the root and offspring of King David has come. Luke is written to a Roman Judge by a Gentile doctor, who is consumed with giving a fairly detailed and accurate account of the person and work of Christ. Mark is a call to the Gentiles showing the immediacy of His work as a servant and the story of the gospel stripped down to just the good news. Each author has taken the events and juxtaposed them topically, thus giving each a particular emphasis to the story.
In Matthew the account is preceded by the healing of the paralytic, the feasting with sinners and the instruction of the Pharisees about the newness of the Kingdom. It's followed by the healing of a man with demons and the Pharisees saying Jesus operated by demonic power.
In Luke the account is preceded by Jesus calming the storm and healing the man possessed by legion. It's followed by the first commission of the disciples.
In the Mark account it's preceded by Jesus healing the man possessed by Legion, and followed by Jesus returning to His hometown to teach.
At first glance Matthew appears to be wildly different, but this is because he is trying to make a point: the Pharisees largely rejected Jesus at every turn and sought to discredit Him. He on the other hand proved His divinity by forgiving sin and raising the dead. In Luke we have a closer narrative of the events, with the teaching of the Pharisees immediately before the healing left out entirely because the focus was on the miraculous ability to heal.
The Bible doesn't contradict itself, nor can it. Instead, every time it appears to it's worth investigating why one author is choosing to emphasize a certain truth and what they are trying to tell us, rather than demand Scripture be at fault or imperfect.
Post Script
I left out reconciling the text because it should be obvious that if you want to reconcile it you should have no problem doing so. However, if your faith is weak, and you are not taking my advice of doubting your doubts, I'll provide you the narrative if you mash all the details together. Luke 8:41-51 is in green, Mark 5:21-37 is in blue, and Matthew is in red.
And when Jesus returned [and] had crossed again in the boat to the other side, a great crowd [was] waiting for him, gathered about him, welcomed him, and he was beside the sea.
"Neither is new wine put into old wineskins. If it is, the skins burst
and the wine is spilled and the skins are destroyed. But new wine is put
into fresh wineskins, and so both are preserved." While he was saying these things to them, behold, there came a man named Jairus, who was a ruler of the synagogue. And seeing him, And
falling at Jesus' feet, he implored him earnestly to come to his house,
for he had an only daughter, about twelve years of age, and she was dying saying, "My little daughter is at the point of death. Come and lay your hands on her, so that she may be made well and live." And he went with him.
And a great crowd followed him and thronged about him.
As Jesus went, the people pressed around him.
And there was a woman who had had a discharge of blood for twelve years, and though she had spent all her living on physicians, all that she had, and had suffered much under many physicians, she could not be healed by anyone, and was no better but rather grew worse.
She had heard the reports about Jesus, came up behind him in the crowd and touched the fringe of his garment, for she said, "If I touch even his garments, I will be made well."
and immediately her discharge of blood ceased and she felt in her body that she was healed of her disease.
And Jesus perceiving in himself that power had gone out from him, immediately turned about in the crowd said, "Who was it that touched me?" When all denied it, Peter said, "Master, the crowds surround you and are pressing in on you!"
And his disciples said to him, "You see the crowd pressing around you, and yet you say, 'Who touched me?'"
But Jesus said, "Someone touched me, for I perceive that power has gone out from me."And he looked around to see who had done it. And when the woman, knowing what had happened to her, saw that she was not hidden, she came in fear and trembling, and
falling down before him declared in the presence of all the people why
she had touched him, and how she had been immediately healed, the whole truth.
And he said to her, "Take heart, Daughter, your faith has made you well; go in peace, and be healed of your disease."
While he was still speaking, someone from the ruler's house came and
said, "Your daughter is dead; do not trouble the Teacher any more." But Jesus on overhearing this answered him, "Do not fear; only believe, and she will be well."
and [he] knelt before Him
"My daughter has just died, but come and lay your hand on her, and she will live."
And when he came to the house, he allowed no one to enter with him,
except Peter and John and James, and the father and mother of the child.
Matthew has only condensed the account and picked up the most important element to the Jew: the raising of the dead.
