Diopter refers to the curvature of a lens. As the diopter increases, the lens becomes thicker and the curvature greater. As the curvature increases, light rays are redirected to fill a greater portion of the viewer's retina which makes the object look bigger.
Power refers to how much larger an object is made to look through a magnifying lens.
Unfortunately, there are two different conventions for computing power that lead to different results. Below is an example of how two power ratings can be achieved for the same 8 diopter lens. Formula A says that the 8-diopter lens makes an object look 2 times bigger, over and above what the unaided eye already sees. Formula B says that the 8-diopter lens makes an object appear 3 times larger than it actually is. Focal length is important when the task requires using tools with the object being viewed.