But with Tensors you can not only express compressive/tensile forces on a plane through a point and not only shearing on that plane but also sort of rotation: curls -> non-symmetric across the diagonal elements. I believe Feynman introduced a nice picture for this: imagine you have a perfectly symmetric nutshell with some thin paper blades attached to it, so it looks like a rotor. Put it right into a stream and fix the position. If the stream is flowing faster around on one side, it will start to spin. That's the curl in that point of the stream.

Edit: for clarity