Both these organizations seem to keep coming up in Railgun/Index, but it's never explained what the difference is. Both of them seem to be some kind of police force. At first I thought Judgment was the "students" whereas Anti-skill was everyone else, but the way Judgment is being used as some kind of swat team when the plot demands it makes that seem strange.
Judgement is a term used to refer to Academy City's student-based disciplinary committee. They are composed of students of varying grade levels and power, and along with Anti-Skill, they are tasked to maintain peace-and-order within the school system. Judgment members can be recognized by the armbands they wear over their right sleeves, which are green with white stripes and which have a shield symbol.
Anti-skill is the term used to refer to Academy City's police and security forces. Unlike Judgment, members of Anti-Skill are ordinary adults who work in the city. One is essentially a volunteer when they join Anti-Skill, and as such is not paid to do Anti-Skill work; however, the city compensates for Anti-Skill members' help in the experiment by giving them various gifts and privileges
It is often assigned to situations that members of Judgment are unable or not allowed to handle, such as engaging terrorists, handling hostage situations, or stabilizing dangerous areas. Judgment is often assigned to assist Anti-Skill in these cases by evacuating civilians or handling crowd control as well as dealing with rogue espers who abuse their powers. It is also shown to assist Judgment with meager work such as directing traffic, helping lost people, and enforcing curfew, and can also be hired for events. Anti-Skill members are also tasked in training Judgment recruits.
Moreover, it is also their duty to keep watch of Judgment, and vice-versa. It is a checks and balances system that should prevent prevent internal corruption in the two organizations