How do you move the vertices of one UV island without moving the same vertices part of the other one? When UV unwrapping, the some UV islands will share vertices with other islands. So when I select a vertex in one island that is shared with another island, both vertexes will be selected.
How do you move one UV island without moving the vertices of another island at the same time?
Switch to face mode in the UV editor, or make sure that UV Sync Selection is disabled, then switch to island mode or set sticky selection mode to Disabled.  All are options on buttons in the menu bar of a UV editor window.