Links in many real-world networks activate and deactivate in correspondence to the sporadic interactions between the elements of the system. The activation patterns may be irregular or bursty and play an important role on the dynamics of processes taking place in the network. Information or disease spreading processes in networks are paradigmatic examples of this situation. Besides burstiness, several correlations may appear in the process of link activation: memory effects imply temporal correlations, but also the existence of communities in the network may mediate the activation patterns of internal an external links. Here we study the competition of topological and temporal correlations in link activation and how they affect the dynamics of systems running on the network. Interestingly, both types of correlations by separate have opposite effects: one (topological) delays the dynamics of processes on the network, while the other (temporal) accelerates it. When they occur together, our results show that the direction and intensity of the final outcome depends on the competition in a non trivial way.