As a Neolithic chief, I would oversee: the construction of, at least, public buildings; the accumulation of and distribution of surplus goods; and conflict resolution. When necessary, I would represent my village in negotiations with nearby villages over boundaries and resources such as water and game.
As for current Neolithic-style chiefs, if any remain, they would have to be in areas isolated enough from modern development for their positions to have substantial merit. In some cases, however, such governance may have been revived after a fallow period due to contact with modernizers. Native American tribal chiefs come to mind, but their role is highly circumscribed by state and national governments.
At this time, every part of the earth is beholden to modern national, state and local governments, yet certain groups surely maintain relative independence even under such circumstances, but they are probably hunter-gatherers because agriculture, which characterized Neolithic society, has brought most “backwards” groups in touch with the wider world through trade of its products. A group would have to be self-sustaining in order to preserve Neolithic patterns, though some villages may still have chiefs despite contact with the wider world. How faithful these “chiefdoms” are to their Neolithic antecedents would vary, but the idea of a chief’s absolute rule, at least in an agricultural setting, may not have survived nationalism.