The identification of the most influential spreaders in networks is important to control and understand the spreading capabilities of the system as well as to ensure an efficient information diffusion such as in rumorlike dynamics. Recent works have suggested that the identification of influential spreaders is not independent of the dynamics being studied. For instance, the key disease spreaders might not necessarily be so important when it comes to analyzing social contagion or rumor propagation. Additionally, it has been shown that different metrics (degree, coreness, etc.) might identify different influential nodes even for the same dynamical processes with diverse degrees of accuracy. In this paper, we investigate how nine centrality measures correlate with the disease and rumor spreading capabilities of the nodes in different synthetic and real-world (both spatial and nonspatial) networks. We also propose a generalization of the random walk accessibility as a new centrality measure and derive analytical expressions for the latter measure for simple network configurations. Our results show that for nonspatial networks, the k-core and degree centralities are the most correlated to epidemic spreading, whereas the average neighborhood degree, the closeness centrality, and accessibility are the most related to rumor dynamics. On the contrary, for spatial networks, the accessibility measure outperforms the rest of the centrality metrics in almost all cases regardless of the kind of dynamics considered. Therefore, an important consequence of our analysis is that previous studies performed in synthetic random networks cannot be generalized to the case of spatial networks.