Early identification of dementia at the stage of mild cognitive impairment (MCI) is crucial for timely diagnosis and intervention of Alzheimer's disease (AD). Although several pioneering studies have been devoted to automated AD diagnosis based on resting-state functional magnetic resonance imaging (rs-fMRI), their performance is somewhat limited due to non-effective mining of spatial-temporal dependency. Besides, few of these existing approaches consider the explicit detection and modeling of discriminative brain regions (i.e., network hubs) that are sensitive to AD progression. In this paper, we propose a unique Spatial-Temporal convolutional-recurrent neural Network (STNet) for automated prediction of AD progression and network hub detection from rs-fMRI time series. Our STNet incorporates the spatial-temporal information mining and AD-related hub detection into an end-to-end deep learning model. Specifically, we first partition rs-fMRI time series into a sequence of overlapping sliding windows. A sequence of convolutional components are then designed to capture the local-to-global spatially-dependent patterns within each sliding window, based on which we are able to identify discriminative hubs and characterize their unique contributions to disease diagnosis. A recurrent component with long short-term memory (LSTM) units is further employed to model the whole-brain temporal dependency from the spatially-dependent pattern sequences, thus capturing the temporal dynamics along time. We evaluate the proposed method on 174 subjects with 563 rs-fMRI scans from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, with results suggesting the effectiveness of our method in both tasks of disease progression prediction and AD-related hub detection.