Neo Illuminati is the alliance of like-minded individual or those that came under circumstance where they has form a bond with Serizawa Yami. Instead of a proper organization, Neo Illuminati is just a group of friend that happen to have special power. And while the Bavarian Illuminati has a set goal that is clearly defined atleast on paper. The set goal for Neo Illuminati is "To purge the world of thing we perceived as offensive." which is a very vague goal.
The Neo Illuminati is the organization that facillitate many conflict around the world either accidentally or to serve a goal that are wholeheartedly petty in comparison to the death toll it cause. It first true action was causing a war between the Fangire and Orpnoch in Val Verde for the purpose of temporarily stop them from pursuing two of it member.
In a sense, Neo Illuminati is even volatile and dangerous compare to the Bavarian Illuminati because they are just a bunch of jerk abusing their power. But contrary to what one might belief they're not hostile to 2nd Division or S.O.N.G. The reason are very obvious considering that several member of Neo Illuminati are directly related or has been infuriated with the Relic User.