Like many other sites, most visitors to our site are not technical people. I'd say the current phpBB profile if more techy compared to profiles offered by BuddyPress, JomSocial, other forums scripts, etc.
Rather than everything being stuck on profile all at once, include content in buttons or tabs. So Groups, Contact, and Statistics would have their own buttons/tabs. This would keep profile simple and clean.
A friend management area on profile page would be great - list of users friends with avatars showing, button for adding user as friend when viewing the profiles of others, ability to remove friend when viewing your own profile.
A list of 10 or so topics and 10 or so relies of user who's profile is being viewed (under a tab/button of course).
An ability to integrate other extensions with button/tabs. For example, if "phpBB Gallery" or "FP's Profile Comments" extensions are installed, these would be located within button/tabs on profile page.
Ability to add a large profile background photo, and maybe have smaller user avatar shown somewhere over this background photo.
I'd love to see this created and would be happy to pay a fee for extension, as I'm sure others would be also happy to do (maybe as a pro extension).