* Fin is the most common and general word for "end", and it can refer to the end of anything: a story, a journey, a project, etc.
* Extrémité is more specific and refers to the literal end of something, like the end of a road, a piece of string, or a line. It can also be used figuratively to mean the extreme point of something, like the extremity of human suffering.
Here are some examples:
* La fin du film (The end of the movie)
* La fin de la semaine (The end of the week)
* L'extrémité du jardin (The end of the garden)
* L'extrémité de la vie (The extremity of life)