show or make something clearer (or more interesting) by using pictures, diagrams or examples.
This diagram illustrates the position of the body’s main organs.
The speaker illustrated his point with an amusing anecdote/example/story.
(1) show or explain through actions
(2) provide evidence that proves something.
(1) First, the instructor demonstrated the move herself; then she told us to try it.
(2) The accident demonstrates the need for improvements to the road.
The accident demonstrates the need for improvements to the road.
= The accident proves or clearly shows the need.
The accident illustrates the dangerous conditions on this road.
= The accident is an example of the dangerous conditions.
You could perhaps use either word in these two sentences, but the meanings are slightly different.