Leave the Office Earlier explains this well:
There is a difference between a responsibility and a task. For example, “participate in team decisions” is a responsibility; “attend team meetings” is a task. “Communicate with customers” is a responsibility; “write monthly ezine is a task.” “Obtain market visibility” is a responsibility; “write article for trade journals” is a task. “Develop media relationships” is a responsibility; “create press releases” is a task.
The distinction hinges on the question, “Why do I do this?” The responsibility is high level, and the task is specific. One responsibility may carry five (or more) associated tasks. If you can eliminate one responsibility through clarification, you may eliminate several tasks. You carry out tasks to fulfill responsibilities.