LARP is an acronym for Live Action Role Playing. It is more a game like Dungeons and Dragons, except it's LIFE SIZE. Instead of sitting around describing the action, it's done in real time and space on location. It is not focused on the medieval period.
The SCA focuses specifically on the MEDIEVAL and EARLY RENAISSANCE period(s) (600-1600 AD), and includes some aspects of role-playing but role-playing is not the primary focus. They have jousting tournaments, cooking and sewing guilds, armor-making workshops, etc. and some folks attend these events "as" a persona that would have existed during that period.
I am interested in that period but I do not have the time to "join" anything. I own a copy of the SCA's Known World Handbook