Actually, they do not report it "differently" in the sense of there being two different stories that contradict each other. As explained by Rabbi Shimshon Rafael Hirsch, or Rabbi Aryeh Levin and others, Genesis one is the description of the creation from the beginning to the end. Genesis 2 starts (paraphrase), When G0d created the universe (as described before), everything was set up. Here are the details of the creation of Adam and what happened immediately after.
It is the same story, but the concentration is on a different set of details. It is like saying "During the sixth day, Adam was created in this specific manner. The following events involving Adam then occurred.".
Note that the start of the "second story" is Bereishis 2:4. The Xian division into "chapters" is mistaken. As Rabbi Shimshon Rafael Hirsc points out, "Aileh" means "These" which points back to what is already said and means expanding on what was said, here are the "products" of what was created (as we described earlier). Now once the world had been created, the important part of the Universe and the part that we need to concentrate on is Adam ...
Later we see that the Torah shows what happened to the world until we get to Noah and the flood. We then follow the descendants of Noah until we get to Avraham. We do not give details of the rest of the world, because that is not what the Torah wants to tell us. A book of American history will not say what is going on in China except as it pertains to the particular part of the history that a specific chapter is dealing with. We do not say that the chapter is pretending that China does not exist.