Both the lithosphere and asthenosphere are part of Earth and are made of similar material. Lithosphere is made up of Earth's outermost layer, the crust, and the uppermost portion of the mantle. In comparison, the asthenosphere is the upper portion of Earth's mantle (which is also the middle layer of Earth). The lithosphere lies over the asthenosphere. In fact, if any material from the asthenosphere were to solidify, it would become part of the lithosphere.

Being closer to the Earth's core, the asthenosphere is a higher temperature as compared to the lithosphere and hence its rocks are plastic and can flow. In comparison, the lithosphere's rocks are more rigid. The asthenosphere is more dense and viscous in comparison to the lithosphere. The lithosphere is comprised of a large number of fragments, each of which is known as the tectonic plate. These tectonic plates are in constant motion and are floating over the plastic material underneath.

Hope this helps. 

Compare and contrast the asthenosphere with the lithosphere.

The asthenosphere is located between 623 and 124 miles below Earth's surface. It is under the lithosphere. This layer is responsible for plate tectonics, due to its high temperatures and the fact that it is plastic and allows seismic wasves to pass through. The lithosphere is the crust and the upper most part of the mantle. It is at the uppermost part of the asthenosphere where the plates of the Earth's crust move, due to the heat and pressure in this layer, the rocks above this layer behave elastically and eventually can break, which causes crustal movement, fault lines, etc. There are convection currents that cause heat to flow from the Earth's interior, towards the surface. This energy is responsible for the movement of the crustal plates. You can envision a bowl of soup with croutons floating on top of it to represent successively, the asthenosphere and the lithosphere.

Compare and contrast the asthenosphere with the lithosphere.

The lithosphere is the uppermost solid part of the mantle.  It is not as solid as the outer layer, the crust, but it is not as liquid as the next layer of the mantle, the asthenosphere, which lies directly beneath it.  Here, the rocks have a quality called plasticity, where they tend to flow more like liquids.  After that, you have the rest of the mantle, which is of a true liquid-like melted rock nature.

Compare Lithosphere vs. Asthenosphere?

The lithosphere is the upper most layer of Earth.  It is comprised of all of the Earth's crust and the very upper portion of the mantle.  That last upper bit of the mantle is different from the rest of the mantle because it is slightly more solid than the mantle below it.  It's far enough away from the Earth's hot core that it solidifies more than the mantle below it.  I like to think of that upper part of the mantle like cheese dip at a party.  If that bowl of warm cheese dip sits still for long enough, the very top layer hardens.  That's the upper part of the mantle. It and the crust form the lithosphere.  

The lithosphere is divided up into giant plates.  Those are Earth's tectonic plates, and they fit together like pieces of a puzzle.  

Underneath the lithosphere's plates is the asthenosphere.  It is made entirely of mantle material.   It is denser than the lithosphere and highly viscous.  That means it flows around very slowly.  Think of it like the consistency of silly putty or warm clay.  As the asthenosphere moves around it pushes and pulls the tectonic plates with it.