The Move/Rename tab only has one function to change the name and/or location of the asset. The Move/Rename screen also shows any affected Destinations.
The best practice before you move or rename anything is to unpublish it to remove it from the live server first. If you don't do this the old version of that asset will forever be orphaned on the live server and users may find that out of date page in a site search leading to confusion and misinformation.
