This is a very interesting question with a very simple answer, which comes with a very complicated qualifier.
First the simple answer.
Neanderthals and dinosaurs fit into the creation story just fine. God spoke and they came into being (the water and skey dinosaurs were created on day five - Gen. 1:20-23; the land dwelling dinosaurs and Neanderthals appeared on the sixth - vv. 24-25). The Bible doesn't say any record of these creatures going extinct (one can theorize this happened at the flood in Gen. 6-8). I've read and heard people say that the Bohemoth and Leviathan in the book of Job are actually dinosaurs, and that this is an indication that they co-existed with people.
That's one theory, and a way people attempt to find unity between archaeological discoveries and the biblical narratives. But, there are many other Christians who do not even attempt to answer this question.
Now for the complicated qualifier.
If I were to give this answer in my seminary class, I will be ridiculed. Why? Because seminary is filled with scholars of the Bible. They understand the Bible in way most Christians don't know much about - the cultural context in which the text was written.
They know that the Bible is a collection of 66 books written by 40+ authors. They know that these books consist of various genres - some are poetry, some are letters, some are history books, some are instructions on building tents, buildings, and artifacts, some are apocalyptic literatures, some are advice columns, and others are biographies.
Within theses diverse genres, you will find ancient styles of writing that was widely accepted as "fact" in the eastern culture, but would be as considered "unreliable testimonies" in the West. For example, the book of John is an accurate biography of Jesus, although it was not written chronologically. That was normal for the original audience, but the accuracy of biography would be put into question in our culture today. This doesn't mean the people of Palestine from two thousand years ago are less intelligent. It was their style literature.
A lot of people miss this simple nuance: The first section of the book of Genesis was written in a different culture with different sets of "reading rules." For example, did you know that this Genesis 1-3 are a re-telling of an ancient creation myth? The original hearers and readers of this book would have been familiar with this. These myths have gods and demigods warring with one another. They create man to serve their selfish agendas, which usually results in more wars (since people were created in the image of these gods). The first section of Genesis was written to hi-light the differences between the Mesopotamian and Egyptian gods from Yahweh (the Hebrew God).
Yahweh creates humanity from a place of peace and love, whereas their neighboring cultures believed that they were created in the image of gods that only looked out for themselves. In the Genesis account, people have dominion over creation, whereas in other cultures, trees, rivers, and animals were gods and goddesses and ranked above humanity. Yahweh loves people (both man and woman) and requires both to be treated with dignity, since his own image is in them. God makes the world good, humanity is at its center, and there is no enslavement. Everything fits together perfectly. God and humans were meant to partner together to make an amazing world.
This is the purpose of the Genesis creation story - to describe who God is, who humanity is, and begin discussion about the possibilities of the beautiful things that can come out of this partnership.
Now, let me ask you a question: If you were the author of Genesis and it was your intent to inform the reader of the character of God and the role of humanity, would you include the details of dinosaurs and Neanderthals? Would you be concerned with the actual age of the earth (which Bishop James Ussher of Ireland in the 17th Century theorized to 6,000 years old)? What about evolution? How many animals fits in the ark?
"How old is the earth?", "Does the bible talk about dinosaurs?" and "Does the bible teach evolution?" are questions that are asked when we assume the first section of Genesis is meant to be read as a historical account. When I asked a rabbi, who studies the Old Testament day in and day out, whether he believed the world was created in six literal or figurative days, his answer was, "I never even considered this question before."
So, to answer the question at hand, I'm sure the dinosaurs and Neanderthals were created by God, but for us to try to fit them into the biblical narratives isn't a wise thing to do. Let's look to the scientists to see what really happened to them.