The selection that is saved when switching between the Weights tool and other transform tools refers to the vertices and bones you have selected in the Weights view. This means that if you select some vertices and bones while using the Weights tool, then switch to a transform tool to manipulate the bones, and then switch back to the Weights tool, the same vertices and bones will still be selected. This allows you to continue adjusting the weights without having to reselect the vertices and bones.
It does not refer to the pie charts or any visual indicators but rather the actual selection of vertices and bones that you are working with.
