We consider the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph G=(V,E) with a positive real weight on each edge. We present a fully dynamic algorithm for this problem in which an update supports either weight increases or weight decreases on a subset of edges incident to a vertex. Our algorithm runs in amortized O(\vstar^2 \cdot \log^3 n) time per update, where n = |V|, and \vstar bounds the number of edges that lie on shortest paths through any single vertex. Our APASP algorithm leads to the same amortized bound for the fully dynamic computation of betweenness centrality (BC), which is a parameter widely used in the analysis of large complex networks. Our method is a generalization and a variant of the fully dynamic algorithm of Demetrescu and Italiano [DI04] for unique shortest path, and it builds on very recent work on decremental APASP [NPR14]. Our algorithm matches the fully dynamic amortized bound in [DI04] for graphs with unique shortest paths, though our method, and especially its analysis, are different.