In my Achivements tab, I was told I had earned the close votes privilege:
But in my profile, I haven't.
Which one is correct?
Most important thing to keep in mind is this: Once you get a privilege at a certain reputation level, it is possible to lose the privilege if you drop below that reputation level. This happens more often than you would think, unfortunately, so it's not uncommon to see this happen.
You currently have 243 reputation points. You need 250 to get the privilege you specified, which is why you don't see the priv. (This is not, however, a bug.)
Also keep in mind that the notifications box doesn't realtime revoke messages (I still have messages from when the global SE wide issue with the Marshall badge gave me three of them, heh).
It is possible for an upvote to have been given for an answer, but then removed by someone or because a user was removed from the system and their upvote was undone because of that).
I do not have in-depth access to see your reputation, but check your "reputation" tab to see if there are any -10s on there.