Procedural activities are sequences of key-steps aimed at achieving specific goals. They are crucial to build intelligent agents able to assist users effectively. In this context, task graphs have emerged as a human-understandable representation of procedural activities, encoding a partial ordering over the key-steps. While previous works generally relied on hand-crafted procedures to extract task graphs from videos, in this paper, we propose an approach based on direct maximum likelihood optimization of edges' weights, which allows gradient-based learning of task graphs and can be naturally plugged into neural network architectures. Experiments on the CaptainCook4D dataset demonstrate the ability of our approach to predict accurate task graphs from the observation of action sequences, with an improvement of +16.7% over previous approaches. Owing to the differentiability of the proposed framework, we also introduce a feature-based approach, aiming to predict task graphs from key-step textual or video embeddings, for which we observe emerging video understanding abilities. Task graphs learned with our approach are also shown to significantly enhance online mistake detection in procedural egocentric videos, achieving notable gains of +19.8% and +7.5% on the Assembly101 and EPIC-Tent datasets. Code for replicating experiments will be publicly released.