This paper demonstrates how OpenMP 4.5 tasks can be used to efficiently overlap computations and MPI communications based on a case-study conducted on multi-core and many-core architectures. It focuses on task granularity, dependencies and priorities, and also identifies some limitations of OpenMP. Results on 64 Skylake nodes show that while 64% of the wall-clock time is spent in MPI communications, 60% of the cores are busy in computations, which is a good result. Indeed, the chosen dataset is small enough to be a challenging case in terms of overlap and thus useful to assess worst-case scenarios in future simulations. Two key features were identified: by using task priority we improved the performance by 5.7% (mainly due to an improved overlap), and with recursive tasks we shortened the execution time by 9.7%. We also illustrate the need to have access to tools for task tracing and task visualization. These tools allowed a fine understanding and a performance increase for this task-based OpenMP+MPI code.