Wolfram Player iOS app: what does the in-app purchase do?

Controls will not function unless you've enabled interactivity. I.e., a Manipulate will render, but you'll be unable to toggle checkboxes, change input fields, press radio buttons, etc. The same will be true for any control in the notebook, even if it does not live in a Manipulate.

Enabling interactivity can be done either by logging into a paid Wolfram Cloud account or by purchasing the in-app purchase. The in-app purchase is one time and will get you the functionality forever. In addition, if the notebook you're opening is a CDF deployed by the Enterprise version of Mathematica, it will work in Player regardless of whether you've enabled interactivity.

Besides what John mentioned above, EnterpriseCDF documents also require no in-app purchase for interactivity. Creating EnterpriseCDF documents requires a Mathematica Enterprise Edition license, which runs ~\$7,500. Someone wishing to distribute full-blown CDFs or Notebooks to a large population could use that option. $10 is a lot for a casual iPhone/iPad user to sink into an iOS app, but it would clearly be a bargain for a college student to get full functionality to all Mathematica notebooks.

Pystab, you're right. This is somewhat complicated; Wolfram needs to produce a YouTube video showing the behavior of sample code on iOS both with and without full interactivity. What excites me is that students (or anyone) can freely author Wolfram code that they can run anywhere. Even without the in-app purchase, that code can perform some rather astonishing computations and visualizations in this app. I had been curious how Wolfram would walk the knife-edge of a compelling free app that would encourage purchases; I think they've done a good job.