vBulletin and FUDForum have very similar functionality and are both implemented as PHP. The main differences are outlined below. More detailed comparison would require trying them both.
Open Source vs Commercial
FUDForum is an open-source product which is well supported with frequent releases and attention to details/quality. vBulletin is commercial (but relatively cheap). I am less convinced than Imre that commercial support is necessarily better than open source, at least I think everyone could agree that the best open-source products are better than the worse commercially supported products, so comparisons should be made on a case-by-case basis.
Databases
FUDForum works with many databases, while vBulletin only works with MySQL.
WYSIWYG
FUDForum does not support WYSIWYG editing. This 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.
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.
Ability to block external search engines
I don't really understand this one. Would being able to include a robots.txt at the root satisfy the requirement?