The difference between assigning and subscribing to tasks is that when you subscribe to a task then you'll receive notifications when something is updated in the task. Subscribing to a task doesn't assign it to the user and you can't run any reports, see the calendar, or anything else related to that. Subscribing is only for getting email and in-app notifications.
This is useful when you want to receive notifications only about one task and not the others. You can disable the project notifications or even all notifications but you still receive notifications about the tasks you are subscribed to.
You can subscribe yourself and other users to tasks. When you're subscribed to a task then you'll see a small eye icon on the task card.
To subscribe to a task click on the task to open the task window. Then click on the "Subscribe" button on the task window side menu.
Click on the user name to subscribe or unsubscribe from the task. You can also see all the users who are subscribed to the task at the top of the task window.
You can see all the tasks you have subscribed to on the "Tasks" page. Set the filter option to "Subscribed to ...".
You can also subscribe to task list. That way you don't have to subscribe to individual tasks and can receive notifications for all tasks in the list.
You'll see a small eye icon in front of the task list name when you're subscribed to it.