In soccer, an Assistant Referee ("AR") is a neutral match official assigned to assist the Referee, generally running along the touch lines (sidelines) and calling offside, nearby fouls, and other infractions, as well as carrying out other duties as the Referee might assign, such as inspecting goal nets before a match. The AR will wear the same uniform as the Referee, and will usually carry the same equipment, plus a flag. Most ARs are also qualified Referees who take turns with their colleagues as to who will serve in which capacity.
A Club Linesman, also called a Volunteer Linesman or simply Linesman, is a person who is a non-playing member of a club or team, a parent or spectator, or other person who is not a qualified match official who is assisting the referee in the absence of a qualified and neutral AR. Because Club Linesmen are not neutral match officials, they are only permitted to raise their flag to indicate that a ball has gone out of touch (out of play over the touch line). The qualified and neutral referee will take on the responsibility of calling offside infractions and determining the direction of throw-in restarts. Though limited, this service is valuable (particularly at competitive youth levels) because the Referee's position often makes it difficult to judge difficult barely-over-the-line situations.
Assistant Referees are sometimes (incorrectly) called Linesmen, probably because it's shorter and easier to say when calling a match in play-by-play, or speaking about a past or upcoming match by pundits and analysts. As such, many people who watch soccer on television also use the incorrect term. As wrong as it is (and it is), virtually no one except Referees seems to care much about the distinction, and so the terms tend to be equivalent in common parlance, particularly at the professional levels.
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It is the same. The term was changed to Assistant Referee to emphasize that the assistant is expected to help the referee and work as a team.