Forum Moderators: open
Regarding the URL, the main reason that you would go with forum.website.com would be if you were hosting the forum on a different server; where forum.website.com would have a different IP address to [www.]website.com.
If you're on the same server, there's not a lot to choose between them. I would keep it simple and go with the subdirectory.
Content is king (as they say) and I have heard that Google (at a conference) said it is better to build more content into an existing web site than it is to build on a seperate domain, effectively spreading the content. Bigger the better.