Calculus is to algebra what algebra is to arithmetic. It enables us to see the big picture of functions and equations in the same way that algebra let us see the big picture of combining numbers.
Click 'Show tangent' and move the point around to see the gradient at any point on the polynomial curve.
Click 'Show segment' to reveal a second point, and compare the gradient of the segment between the two to your tangent line. Notice how as you move the two points closer and closer together, the segment's gradient approaches that of the tangent line.
Finally, click 'Show gradient function' to see how a graph of the gradient ends up looking like a (simpler) polynomial graph of its own.

Differentiation is a way of finding the gradient between two points when they are no distance apart, by considering what happens when they are very close and then observing what limit they are approaching.
A good example of when this is useful is displacement, velocity and acceleration - velocity is the differential of displacement (the rate at which displacement changes over time), and acceleration is the differential of velocity.