I have integrated phpbb with wordpress; it took a decent bit of work, but I'm still not happy with the result.
The main problem is login integration. There are some problems where it doesn't log into wp, when you're logged into the forum. And, most people place the website/blog at the front of the site, so having the logins on the phpbb side doesn't make sense, and makes it more awkward for the user.
Note: you can't dynamically have the same header/ layout as your wordpress site without creating a bridge between the two, so you'll be required to use copy the html output of anything from wp you want on your phpbb forum. Then copy the css over, as well to the phpbb stylesheets.
Another problem: wordpress has a lot of plugins which can quickly progress your site, however, having the login going through phpbb can restrict this and also cause confusion for the user.
I am currently using phpbb, but the integration is not ideal at all. I'd love it if someone could create a bridge so that wordpress login comes first and phpbb is the subordinate (a lot more practical).