Forums = more page views. Forums = more time spent by user. Forums = less adsense revenue. Forums = more content generation. Forums = More SEO effort.(If you don;t SEo your forums, the content generated by users dont really help since SE's dont see them)
Forums is a good one way links...Good source of backlinks.. Drives also traffic to a site where the members visit the site in their signature links...
I would honestly say it depends on the site. A site about acne treatment most likely wouldn't need a forum, thought it could be benifical to it. Now, a site related to a popular game would be another thing all together, since it would create a lot of discussion a forum would be a great idea. Forums tend to drive down CTR greatly due to so many pageviews in so little time, I don't use PPC in my forums, I only use CPA.
Forums have a great advantage not mentioned here yet. If you want to get your website crawled continually, then adding a forum will do just that. As mentioned above you must include good seo settings to achieve this.
frequent added content mean frequent spider crawling. but forum only works if you find a good topic, seo it the way it should and run it smoothly so traffic could grow.