Tasks and activities are functions you may be able to enter time against in your timesheet.
Tasks are associated with projects, while activities can be used either alone or in conjunction with projects and tasks. That is, you can be allowed to enter time against both an activity and a task.
In systems that track time against projects, activities can be used for functions that are common to multiple projects. For example, a consulting company carries out budgetting, design, and testing for all internal projects. They could choose to make each of these functions an activity. In that case, each of these functions would be available for each project, without project managers having to create a task for each, under each project.