The history and character of the Twilight world are intact in Luminosity up to the point where the story begins, with one exception: Bella's a rational self-awareness-junkie with a penchant for writing down everything that crosses her mind in a notebook. She maintains many of the traits and dispositions of canon - and Luminosity is a Bella/Edward story - but she's a distinctly different character.
The first several sections of Luminosity are very similar to canon in terms of the events that occur, although aspects of Bella's internal monologue differ strikingly.
A few thousand words in, the plot is unrecognizeable.
- TV Tropes page here (warning, potential spoilers).