Does anybody how this works or is this feature not available?
Activities do show up there and you can also add the "project" column in the list view if you'd like. For tasks on the project we're rolling out a list view very very soon!
I'm having the same issue. Seems like at first, some assigned activities with due dates did show in the Activities tab, but they don't anymore.
The activities in projects are still in your activity list. You can add the "project" column like shown here to order them out in your list :) But, tasks are not in this list. Very soon we'll be releasing a task list view for this.
I didn't realize when I posted the question that Tasks were not treated like Activities... a misunderstanding that was compounded because there is an Activity of type Task. I'm still not clear on what the purpose of the non-activity Task is supposed to be or how to use it efficiently.
With this new form of task it's optional to set a due date/time and optional to add an owner. So it's used for things that don't have to be in your calendar most likely.