What does it mean to adapt a story and what does it mean to illustrate a story?

To adapt a story means to change it for various purposes. An adaptation can be turning a novel into a play. Some works of art are adapted for kids such as Gulliver's Travels.

To illustrate means not to change the story, but add drawings that depict what is happening in the story. An example of this can be the Lord of the Rings books which were famously illustrated by Alan Lee.