phpBB is much more powerful, so, if you're going to build forum like this one, go with phpBB. bbPress is simple solution, though good!

Only advantage that bbPress has in this comparison, is that it's easy to integrate to WordPress.

If the forum is the main reason for the site, I would allways go with phpBB.
(if I had to choose from these two)

bbPress or Lightweight is a plug-in and used to creating the forums on top of Word Press website. Also bbPress consists of custom posts, layout designs, widgets and CSS code.
PhpBB is a web application, open source of free forum software and used as a discussion, message boards by which users keep in touch with different communities of peoples. Also PhpBB has most complicated characteristics.

You have already answered it yourself

phpBB is much more powerful and is a standalone software, while bbPress is a plugin for wordpress that keeps the forum to a very minimum. I personally like bbPress because all my websites run on wordpress.

Yes phpBB is more powerful but make sure it must be powered by "vbulletin" software to avoid the spammers in future.

I'd vote for bbPress. It has really developed into a pretty advanced platform, it is lightweight and it can fully integrate with WP site - what not to love. Anyhow, here is a good comparison of the two for those interested to read in more detail