A lot of forums don't use them at the moment, but I can see it becoming a big thing for forums in the near future, especially phpBB and MyBB ones since it's so easy to implement.

Forum-specific Moderators' jobs are to manage only 1 section of your forum, or maybe a couple of sections. The main thing to remember is that they don't manage your whole forum. This is an advantage, because it means more moderating is going on since that Moderator doesn't have to watch over the whole forum, and it also means that the Moderator can take the time to learn more about the specific topic of the forum they are moderating.

This helps because then they can post often and post accurately in their section, and also moderate it to a decent extent. I think that forum moderators will become a big thing in the future and replace global or full moderators, because lets be honest, who would do their job better? Someone who is focused on one specific section of the forum, or someone who has a whole forum to manage?

There are exceptions to this though, of course. Massive forums (1million+ posts) are better off with global moderators, otherwise they'd have way too many staff members to manage effectively. But smaller forums (Less than a million posts) should adapt this thinking and turn the right way, to enable people to do their jobs better and make it a lot easier for them.

I like forum-specific moderators better too because that means they would be more focused to maintain the sections they are responsible for, organize discussions, encourage members to post or start thread, and make sure that things run properly there.

I am a part of a huge anime/manga community, with thousands of members and millions of posts. There we have lots of forum-specific moderators, maybe around 30 members, and to be honest they are very helpful. They are the closest to the members, know what happens most in the section, thus are very helpful in terms of section development. With a good system of work delegation, they are the front, most important part of the staff team as to come with new ideas and carry the execution of site plans and events. Not to mention the enforcement of the rules and management of spams posts or troublesome members.

It also depends on the activity of each staff member, actually, but in general it works.

So exactly your opposite, I would advice to have more forum-specified moderators in huge forums like that with several global moderators to back them up, because there is no way global moderators and administrators could keep everything in check.

Just another point of view.