Illustrate
show or make something clearer (or more interesting) by using pictures, diagrams or examples.

EXAMPLES: 

This diagram illustrates the position of the body’s main organs.

The speaker illustrated his point with an amusing anecdote/example/story.

Demonstrate
(1) show or explain through actions
(2) provide evidence that proves something.

EXAMPLES:

(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.

COMPARE:

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.