As sasha mentions, your problem is actually a generalization of the usual graph isomorphism. To put it differently, graph isomorphism is a special case of your problem, in which all weights are the same. Therefore your problem can only be more difficult.
On the other hand, it is easy to reduce your problem to the usual graph isomorphism. Assuming that only edges have weights, the idea is to split each edge into a path of length two, and to attach to the middle node a clique whose size depends on the weight (we only need that different weights have different clique sizes, and that the cliques be large enough). So your problem is GI-complete, i.e., equivalent to graph isomorphism.