Recent availability of geo-localized data capturing individual human activity together with the statistical data on international migration opened up unprecedented opportunities for a study on global mobility. In this paper we consider it from the perspective of a multi-layer complex network, built using a combination of three datasets: Twitter, Flickr and official migration data. Those datasets provide different but equally important insights on the global mobility: while the first two highlight short-term visits of people from one country to another, the last one - migration - shows the long-term mobility perspective, when people relocate for good. And the main purpose of the paper is to emphasize importance of this multi-layer approach capturing both aspects of human mobility at the same time. So we start from a comparative study of the network layers, comparing short- and long- term mobility through the statistical properties of the corresponding networks, such as the parameters of their degree centrality distributions or parameters of the corresponding gravity model being fit to the network. We also focus on the differences in country ranking by their short- and long-term attractiveness, discussing the most noticeable outliers. Finally, we apply this multi-layered human mobility network to infer the structure of the global society through a community detection approach and demonstrate that consideration of mobility from a multi-layer perspective can reveal important global spatial patterns in a way more consistent with other available relevant sources of international connections, in comparison to the spatial structure inferred from each network layer taken separately.