IMO:
phpBB, IPS, IB, and the rest are basically all the same when you average out all the pros, cons, likes, and dislikes.
XenForo is different. It feels different. And to be honest I don't think it can be classified with any other forum solution, free or otherwise.
It's hard to put your finger on the why, and I bet there are many people here who can feel it too.
I have to disagree to a degree. wWhilst they all have a common solution, they have different markets, and, thus adapt different to these markets. For example, with vBulletin, the software seems to be leaning over into the corporate side. This is shown through what I see, the unintuitive interface, and wierd colours.
XenForo is more user-based, placing an emphasis on user interface, and interoperability. The software is using modern web technologies, such as HTML5, in parapitem closure, to create an ideal solution, which is: easy to use, aesthetically pleasing, ergonomically pleasing, whilst not sacrificing function.
Additionally, the software is dependant on well-maintained external sources, such as TinyMCE. I have yet to see a forum using TinyMCE as their main RTE mechanism in posting content.
I think XenForo can be classed as a forum software, and although the same genus as vBulletin, IP.B and the like, it builds on the current data available.