first, the list at the bottom you refer to is unread posts… its show, at least in our provided themes, on the main forum page to all users but admins… you can of course, turn it on for admins too (they usually use the admin bar and dont need it)… but its all part of core and not a plugin… and the list will vary from user to user based on new posts from their last visit… users may get from 0 to the defined limit in template function depending on new posts since their last visit…
the new template tag you referred to that is part of the template tags plugin would display the most recent topics to all users outside of your forum… though you could do it there too… the template tag displays the last X new topics from your site and will be the same for every user (with exception of handling permissions to view)…