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.
Both are layers of the Earth. The lithosphere is crust and the upper mantle of the Earth, what is typically considered the tectonic plates. Below that, lies the asthenosphere, which is the remainder of the mantle, or the lower mantle, and what the plates "float" on as they move and shift.
The main difference between the two is their distance from the surface. Athenosphere is between 620 and 125 miles below the surface of the Earth, which is under the lithosphere. The lithosphere is found to be the uppermost solid part of the mantle. The rocks at this point are more “liquid”.
Some similarities are that they are both close to tectonic plates, allowing for the passage of p-waves. In addition, they both contain lots of Silicate.