Computer-aided prediction of dementia status (e.g., clinical scores of cognitive tests) from brain MRI is of great clinical value, as it can help assess pathological stage and predict disease progression. Existing learning-based approaches typically preselect dementia-sensitive regions from the whole-brain MRI for feature extraction and prediction model construction, which might be sub-optimal due to potential heterogeneities between different steps. Also, based on anatomical prior knowledge (e.g., brain atlas) and time-consuming nonlinear registration, these preselected brain regions are usually the same across all subjects, ignoring their individual specificities in dementia progression. In this paper, we propose a multi-task weakly-supervised attention network (MWAN) to jointly predict multiple clinical scores from the baseline MRI data, by explicitly considering individual specificities of different subjects. Leveraging a fully-trainable dementia attention block, our MWAN method can automatically identify subject-specific discriminative locations from the whole-brain MRI for end-to-end feature learning and multi-task regression. We evaluated our MWAN method by cross-validation on two public datasets (i.e., ADNI-1 and ADNI-2). Experimental results demonstrate that the proposed method performs well in both the tasks of clinical score prediction and weakly-supervised discriminative localization in brain MR images.