Shale is a clastic sedimentary rock composed of silt and clay grains. It is usually rather soft and brittle. Most shale is impermeable which means it doesn't allow water to move through it.
Slate is a metamorphic rock.
It used to be shale but it has changed due to metamorphic activity like heat and pressure. Slate is usually very strong and impermeable. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.