The primary difference between the lithosphere and the asthenosphere is that the rocks of the asthenosphere are more ductile than those in the lithosphere, which means that they can flow like silly putty, whereas the rocks in the lithosphere just bend or break. That is why the lithosphere is made up of tectonic plates.

The asthenosphere is hotter and denser than the lithosphere and is under much more pressure. That is part of why the rocks in the asthenosphere can flow. They also have different minerals. There is a much wider variety of minerals in the lithosphere, while in the asthenosphere, there are mostly iron and magnesium silicates like olivine (what jewelers call peridot).