The interface looks identical, the only difference is that the little thing at the bottom of each post that says "report" is replaced by another button that says "remove" and it then asks for the reason we are removing it. We cannot access people's accounts.
Super moderators can - Remove questions, updates to questions, answers, comments, comments to Reality Check profiles and comments to articles/stories. To do anything else we would have to go through the same process as a normal user.
Moderators can - Remove answers, comments, comments to reality check profiles, and comments to articles/stories.
I do not think forum mods are normally granted those powers. Those powers are commonly restricted to forum administration who are employed by the owning organisation. If there is an issue regarding someones account, the moderators normally contact forum administration who deal with the issues, and that end the moderators involvement in the process. This is from what I understand of regular forum structure.
Moderators can delete comments from Answers. and Answers.
Supermoderators can delete questions, can delete comments from Articles, comments from Answers, and Answers themselves.
The interface is basically the same except where regular users see "Report" next to an answer or question or comment, the mods see "Remove" when they click it, a comment is removed. If they click it next to a Question or Answer, then a drop down box of possible reasons appears. They select one, hit "Remove" again, and it's done.
We cannot access peoples accounts or private messages, though for some reason, rumours that there's a bug in the system being exploited by some mod/supermods continue to do the rounds.
Unlikely, at it would be a total invasion of privacy. Admin, I'm sure, however can access a user's profile, delete the 'about me' if it's deemed neccesary, remove pictures and the like. I've heard of being able to look into private messages, but I don't know if there's any proof to support this claim.
Moderators can remove answers and comments on answers. They cannot remove questions, they can only flag/report them (I'm not sure if they are treated differently from a normal user reporting a question). They do not have access to user accounts or anything like that.
I know this because I am a moderator.
I do not know if there are "super mods" or if they are simply administrators. I would imagine the latter at least can do anything they like on this site since it is their site.