vBulletin and FUDForum have very similar functionality and are both implemented as PHP. They appear to both be very similar:
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(The later compares an old version of FUDForum). The main differences are outlined below. More detailed comparison would require trying them both.
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FUDForum does not support true WYSIWYG editing. This , although it does have a "Rich Text" editor which makes creating messages easier. Adding true WYSIWYG does not appear to be a priority for the developers, although there are some pointers to how to add this functionality yourself.
http://fudforum.org/forum/index.php?t=msg&goto=162516&
Bi-directonal e-mail support
I saw no indication that vBulletin support this functionality, in fact some indication it does not:
http://www.vbulletin.com/forum/showthread.php?300725-Mailing-List-integration
Again there are some hints that this can be added yourself:
http://www.vbulletin.org/forum/showthread.php?t=151222&highlight=mailing+list
Integration with common identity stores.
FudForum has a PHP "hook" for integration with other login systems. We would need to look at this in more detail to understand how easy it is to use: http://cvs.prohost.org/index.php/Login_integration
I am not sure what integration vBulletin provides. There appears to be discussion of an LDAP plugin but it is not clear what level of support this has?
http://www.vbulletin.org/forum/showthread.php?t=231909
Ability to block external search engines
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Perhaps there is an FUDForum->vBulletin importer. On the other hand the links in this post all appear to be broken (commercial support at work?).
http://www.vbulletin.com/forum/showthread.php?151319-FUDForum-Import
There appears to be a script for vBulletin->FUDForum. Unclear if it works:
http://fudforum.org/forum/index.php?t=msg&goto=35104&
Detecting duplicate threads
FUDForum does not support this. I wonder how well it works in vBulletin?