Great question! Well, as you can see here I am using a Joomla (with the Joomla Fireboard Component to run the forum). I wouldn't recommend it though. As you can see, the forum looks nice and is functioning just fine. So it's impt that it is stable software and is running alright. The downside to the forum is that it is not SEO friendly at all. The links are long and akward, and all of the SEO components (that can be used for free and downloaded online) have so many problems, they slow down the site and in the end cause big headaches. So after many issues, I gave up on that and put it back to the default long URLS. SEO is a big deal to me but the forum staying online is more important.

So....I haven't used an alternate script for a while (like phpbb) but if you want something more on the social network/forum type of side, definitely go for: Dolphin software. It is opensource (free) and has beautiful features. I installed it before and was completely impressed by the massive amount of features and income potential. It has a built in affiliate/referral program. You can use it or leave the site as it is (free to join). Here is a link:
http://www.boonex.com/products/dolphin/ You can also checkout (and tryout the demos) for different forum softwares here:
http://www.opensourcecms.com/ (all free)