weight balanced binary tree vs height balanced binary tree
The way to answer your question is to find the peer-reviewed papers that first described the trees (or that later chose to reference them) and read their explanations of why the authors felt the need to invent a new tree. 

Generally speaking, weight-balanced trees have the property that restructuring operations on a node with $n$ descendants happen every $O(1/n)$ operations. This is good because sometimes these trees are used in an augmented way so that restructuring operations take $\omega(1)$ time. In some height-balanced trees, restructuring operations can happen every operation, while in others they happen every $2^{-O(\lg n)}$ operations, which is slightly weaker than $O(1/n)$.