Tickets are typically used for customer related issues and requests or used for technical bugs by software developers. You can group them into categories and use the statuses to manage them such as 'reviewing', 'on hold', 'testing', 'closed', etc., and reassign them through the process of fixing the bugs.
Tasks are used for getting things done and assigning items to people to meet deadlines, and therefore typically have due dates. For more complicated tasks, you can use subtasks to break up the work between phases and / or people. You can manage tasks using the Gantt or Calendar views as well.