Not much but it is not as simple as it may seem.
What you call the Crust and is generally referred to as Lithosphere includes the top part of the Earth’s Mantle. The Asthenosphere excludes that top part as well as the deeper part of the Mantle.
The reason for this name change and overlapping of the Mantle with its closest neighbours are: the density and the chemistry of the various Earth’s sections the etymology of the very word Asthenosphere as well as in its physical function.
Lithosphere Mantle and Nucleus are defined and separated by different percentages of three basic elements Silica Iron and Nickel. But we are not concerned here we their actual composition.
As for the etymology (pronounced stenos) in Greek means resistance rigidity resilience… If you put an in front of it (making it astenos) you have the opposite of the word that is: week fluid flexible.
Therefore the asthenosphere is fluid. It cannot include the top section of the Mantle (that is already rigid) nor its bottom part near the Nucleus that is nearly solid (density = 9 to 12 g/cm3) because of the huge pressure.
As for the physical aspect that explains the reason for this rather confusing mix of genders we must consider Plate Tectonics.
The very notion of Asthenosphere is essential to its mechanism. Plates float and move about as carried by convective currents within the molten magma constituting the Earth’s Mantle. Nothing like this could happen in a solid media. Plates could not possibly drift on a rigid solidified lava block.
For that reason the crest of the Mantle was excluded from the Asthenosphere and included in the Crust in spite of its chemical composition.