Awesomebb, like every other "software" Forumotion has on their hosting is usually an edit of phpBB, the layout looks different but it functions similar to phpBB. So it's essentially just a reskinned phpBB3.
You're welcome, if you're looking for free forum hosting I would go with Jcink over Forumotion. If you decide you want to self host, you can download a database backup of your forum on Jcink. You can't do this on Forumotion.
I always go self hosted. Just been looking around at old free forum hosts to see which are still around and come across Forumotion offering AwesomeBB. I used to frequent free host support forums back in the mid to late 00's unfortunately their all offline now.