While reading a commentary about Daniel 2, I came across the author's idea that Nebuchadnezzar was not real happy with his wise men. It may be that as a young ruler, he was not keen on having his father's leftover counselors attempting to tell him what to do. Along with their claims to be messengers of the gods, he was probably frustrated with them. His dream, then, was just the opportunity to test their claims and abilities. Were they able to do the impossible by telling him his dream and interpreting it? No, they weren't. And Nebuchadnezzar was unwilling to let these self-proclaimed "wise" men get away with their claims of superiority. It now time to put up or shut up.
In contrast to the other wise men who were so filled with terror that they had no plans and had already been cut off from any additional time, Daniel, who had not been a part of the king's frustration with his older counselors, was granted his request. It is possible that Daniel's calm assurance that his God was able to help him somehow impressed the king that here was honesty and integrity quite in contrast to his fawning, older counselors.
John F. Walvoord, Daniel: The Key to Prophetic Revelation, (Chicago: Moody Press, 1971), 54.
This is what makes the difference between those who follow false gods and those who know the Creator God of the Bible. Just as it happened with the prophets of Baal on Mount Carmel and the Pharaoh's magicians during the plagues, there is a big difference between the true follower of God and those who follow Satan. And that will be quite evident in the attitude with which they handle impossible situations like this one.