Both allow for creativity, just in different ways. Most adaptations are based off of what is already public domain (Hamlet, Romeo and Juliet, Headless Horseman, etc…), while fan-fics tend to keep the original characters and settings of the original work; but maybe there’s an additional character, or a character has a different personality, or there’s a nuance in the plot.
Adaptations can also be the original work just in a different medium (movie into book, book into movie, movie into comic, etc…).
Fan-fics can evolve into their own work (very few do), which mostly happens when authors draw inspiration from the original work but have completely different settings, plot, and characters, but that doesn’t automatically make it an adaptation (think Fifty Shades of Grey).
An example of fan-fiction: An Avengers fan-fic, but Tony Stark has a daughter.
An example of adaptation: West Side Story is a more modern adaptation of Romeo and Juliet.