I just downloaded phpBB and Invision Board, and have been trying them both out. They both installed extremely easy on my Mac and were up and running in literally 5 minutes. (8min if you consider the time it took to set up a virtualhost). Here are a few reasons why I'm leaning towards Invision:
1) The admin panel is much more comprehensive than phpBB.
2) The templates are easier to edit. (directly in the admin panel)
3) Calendar of events
4) Support for "Fast Reply"
5) The layout of the navigational aides is nicer in Invision than in phpBB. IMHO
6) Invision offers cheap web hosting for your forum, or for your entire site. Fairly comprehensive package (at a glance anyway)
I haven't tried customizing any of the Invision scripts yet, but one of the things I liked about phpBB, was the ease of changing the "jump menu" (for example). Not to mention that phpBB is open source... which is always something I ike to support.
Anyway, that's my 2ยข. I'd be interested in hearing other stories in favor of phpBB. I've seen a few nicely customized phpBB forums (BuzzLY's comes to mind) and I'd like to know why others might pick phpBB over Invision Boards. (I'm leaving vBulletin out of this because of the cost issue, and I haven't tried other boards yet)