Q. Luke tells us how Jesus went to the house of Jairus, whose daughter was “dying” (at first) but apparently “dead” when Jesus arrived. Jesus said, “She’s not dead but asleep.” But Luke says that when he took her hand, “Her spirit returned.” Then Jesus told the girl’s parents not to tell anyone what had happened. So was the girl dead or asleep? Why did Jesus tell them she was “not dead”? And why did he tell the girl’s parents not to tell anyone what had happened, when he had just instructed the man who’d been freed from the legion of demons to tell people how much God had done for him?
As for Jairus’s daughter being dead or not, I think the key to what Jesus meant is that a common expression for death in ancient cultures was “falling asleep.” We see this in several places in the New Testament, for example, in 1 Thessalonians, “We do not want you to be uninformed, brothers and sisters, about those who are asleep, that you may not grieve as others do who have no hope,” and in 1 Corinthians, “Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep.” We also see this in the Old Testament; a common way to describe someone dying is to say that they “went to sleep with their ancestors.” While the Old Testament examples are in Hebrew and the New Testament ones are in Greek, the image would also have been used in the Palestinian Jewish culture of Jesus, where Aramaic was spoken, so we can be confident that the Greek-language New Testament writings are preserving Jesus’ expression accurately.
Now “fallen asleep” has the connotation of potentially waking up; “died” is more final. So I think Jesus meant that the wailing and mourning were not appropriate in any event because the girl already had the hope of life after death, and moreover in this case she was just about to be raised from the dead as a proclamation of the kingdom that Jesus came to bring. So the main point really is, “Stop wailing,” or, “Do not grieve as those do who have no hope.”
As for why the man freed from the legion of demons was told to tell all his friends what happened, while this girl’s parents were told not to tell anybody, I think this has something to do with is sometimes referred to as the “Messianic secret.” Jesus couldn’t let his own people know too soon who he was, or that would provoke deadly opposition from their leaders before his purposes on earth had been completed. But there wasn’t that risk among Gentiles; instead, the proclamation among them prepared the way for further ministry to them by Jesus (he later went back across the lake and this time the people wanted to see him for teaching and healing) and for proclamation of the good news to them by the apostles after the resurrection.
Good Question 
Interesting about how Jesus wanted one person to tell the others about him and not the others. Helpful insight.