weight balanced binary tree vs height balanced binary tree What are advantages and disadvantages of weight balanced binary tree over the 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 $O(1/n)$.