Different weighted scale-free networks show weight-topology correlations. A clear signature of these correlations is given by the non-linear scaling of the node strength with its connectivity. In this paper, we show that networks with and without weight-topology correlations can emerge from the same simple growth dynamics of the nodes connectivities and of the links weights. A weighted fitness network is introduced in which both nodes and links are assigned intrinsic fitness. This model can show a local dependence of the weight-topology correlations and can undergo a phase transition to a state in which the network is dominated by few links which acquire a finite fraction of the total weight of the network.