At What Scenario will use Activity Chaining and Swimlane?

In simple terms, activity chain helps simulate wizard like functionality. Swimlane help you execute in a particular context and also visually separate task/activities per role/some times functionality.

Please share the use case if you have any. I you are just exploring/learning Appian, I suggest to use Appian community edition try out both features and in combination.

Swimlanes are useful for organizing your process according to which actors (such as Student, Teacher, Staff, Computer, Google AI, etc.) are doing which steps.  This is especially helpful if you have many processes that change hands frequently, such as approvals, reviews, sign-offs.  You can also use swimlanes to determine who gets all the User Input Tasks inside the lane, so you don't have to configure each individual one.

Activity Chaining causes the form on the next User Input Task to appear immediately, provided that the two tasks are supposed to be assigned to the same user.  When you need to show multiple completely different interfaces, and especially when you need to do some process model processing in between, this provides for a better experience because the user doesn't have to go all the way back to Tasks to find the next screen over and over.  They can just get the next part automatically.  Plus it also helps to make sure no one skips a step.  If the processing in between takes more than 30 seconds, activity chaining can be a bad experience instead.  And if there's a task someone else has to complete in between, the activity chaining won't wait.