Ultimately it has to show the same data unless the site is completely discontinuing the service (in which case you are up a creek in either scenario). Even in the face of a total redesign you're usually just adjusting a few signatures and maybe tweaking the traversal tree.

You can run into problems when they remove things like chronological listing view in favor of some "algorithm" that you can't control, but even that just adds a little work on the back end to detect duplicates and an asterisk that some of the content will be missed.

There can also be an arms race where the site developers will start trying all kinds of crazy techniques to make scraping harder. Ironically this often happens after they revoke API access, meaning users don't have an alternative except to participate in the arms race.