Thanks to Maurice's very helpful
embed_phorum mod I've been able to successfully integrate Phorum with Drupal. My code is available as a Drupal module called
phorum_integrate.
The approach I'm using is for Drupal to do the user authentication and Phorum to act as a slave. User IDs are shared between Drupal and Phorum. This means it's not suitable for integrating an already in use Phorum database, only a fresh install, because otherwise the user IDs might not match. For that case you'd need a new table to map between user IDs: I guess you could use email addresses to create the initial mapping.
The Drupal core forum is not the best, so being able to integrate Phorum is a positive step, I think. It would be good to have them integrated more closely, so for instance they could use the same BBcode plugin and other content creation tools. A Phorum module which delegated to Drupal's input filtering would be one way to get there.
Also it would be nice to be able to present content from the forum on other pages (a 'latest posts' block or the like). All other suggestions welcome.