The lithosphere is solid rock from the crust and upper mantle, while the biosphere is living and dead organic matter.
The lithosphere is the crust and upper mantle of a planet, including all the solid matter from moutains to valleys to tectonic plates underneath. In Earth the lithospheric mantle is brittle and hard, almost like the crust, though chemically distinct.
The biosphere is the life and ecology of a planet. It is not a distinct area, but rather a collection of areas, including parts of the atmosphere, lithosphere and hydrosphere, where organisms live and dead organic matter rests.
There is a definite overlap of the lithosphere and biosphere, such as oil and coal buried in the ground, or fossilisation, where lithospheric processes bury and preserve parts of the biosphere. However, they are distinct.