In THIS WinXP system, when I move desktop icons slightly, it's like they are on elastic bands, and they snap back into place. In another similar WinXP system of mine on a different computer, the icons stay put in any higgeldy-piggeldy position I drag them. The settings on the desktop menu for 'Arrange icons by...' are identical for the two WinXP systems. How might I fix the faulty (the latter) one ?

Right click on an "empty" area on the Desktop. Under "Arrange Icons By", deselect "Align to grid".