Accurate survival prediction is crucial for development of precision cancer medicine, creating the need for new sources of prognostic information. Recently, there has been significant interest in exploiting routinely collected clinical and medical imaging data to discover new prognostic markers in multiple cancer types. However, most of the previous studies focus on individual data modalities alone and do not make use of recent advances in machine learning for survival prediction. We present Deep-CR MTLR -- a novel machine learning approach for accurate cancer survival prediction from multi-modal clinical and imaging data in the presence of competing risks based on neural networks and an extension of the multi-task logistic regression framework. We demonstrate improved prognostic performance of the multi-modal approach over single modality predictors in a cohort of 2552 head and neck cancer patients, particularly for cancer specific survival, where our approach achieves 2-year AUROC of 0.774 and $C$-index of 0.788.