The advantages are numerous and extremely well-documented around here, but perhaps the Daily Statistics is the best way to summarise what XenForo has done for us:
Activity all over the site - not just on the forums - is up, not least because of the infinitely more accessible (and fun!) user interface XenForo gives us, as well as a collection of new features and the impending release of the new Pokémon games on the 12th.
We've got a few bespoke features, including the Trainer Card Maker, a graphics generator used to generate "trainer cards", like this:
This feature is incredibly popular and generally causes most of the traffic we get.
We've also got a new feature, the Creative Corner, which is an integrated artwork and written work gallery system that is barebones at the moment, but will gradually be expanded to provide a gallery with particular emphasis on artists and writers.
We also have a Cryptidex, which is our brand of Pokédex that's basically a jazzed up forum. It has proved popular, though, which proves you don't always need fancy custom applications to get the job done.
Now, despite the site being not exactly a default XenForo install, we don't actually use that many add-ons, which is really a testament to how featureful XenForo 1.2 is:
We've got 7,800 discussions, 180,750 messages, and 44,000 members, as well as over 37,000 trainer cards and 143 creative works created since September 4th, so although we're not nearly gunning for any big forum lists, we have a small, committed, and active community. And now we're running the finest web platform in the world, so it's only upwards from here.