There are several differences. First, in the Lilith and Adam story, Lilith is an independent creation, not derived from Adam like Eve. Second, Lilith's relationship with Adam was one of equality at the start, but it turned contentious when she didn't want to be beneath him. In the Adam and Eve story, Eve is more of a helpmate to Adam from the get - go. Third, the Lilith and Adam story often has connotations of Lilith's defiance and her subsequent exile, which is not part of the basic Adam and Eve narrative.
Well, the main difference lies in the nature of the female characters. Lilith was created simultaneously with Adam and was his equal. She was a strong - willed figure who refused to be dominated. In contrast, Eve was created from Adam's rib in the Adam and Eve story. Their relationship was more hierarchical with Adam being in a more dominant position. Also, the Lilith and Adam story has a more complex and somewhat darker undertone with Lilith's departure and her association with certain 'other' aspects, while the Adam and Eve story is more about the fall from grace in a different way.
The Lilith and Adam story differs from the Adam and Eve story mainly in the role of the female figure. In the Adam and Eve story, Eve is created from Adam's rib and is seen as secondary to him. But Lilith was created equal to Adam. Also, Lilith refused to submit to Adam, which is a very different dynamic compared to the more submissive Eve in the other story.