The biggest disadvantage of HTTP Authentication, from a user's point of view, is probably the fact that you get an ugly looking dialog box, and not a nice form integrated into your website.
You also cannot include any link to a "register" form, or some help, nor some "I've forgotten my password" information.
For some kind of back office, maybe http authentication is OK ; but I have some doubts about its usage for some public front office.
Another inconvenient is that there is no "auto-logout" functionnality, with HTTP Authentication : with sessions, the session expires after some time, or the cookie is automatically deleted when the user closes his browser... But not with HTTP Authentication ; so, on this point, HTTP Authentication seems less secure.