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There are two applications for creating stories. Both have almost identical capabilities; they differ only in the interface you use to create a story.
Twine provides a visual interface, where you can see a map of your story and easily visualize connections between passages. It includes an integrated text editor for editing passages. Most people will want to use Twine.
twee is meant to be run from the command line, after you write specially-formatted text files with the text editor of your choice. If you already are comfortable with a specific text editor or development environment, twee is a better choice for you. Creating stories with twee can also be automated or included in other scripts much more easily than Twine can.