If the symlink targets are directories, you need to add the -T flag to the mv command, otherwise it moves the new symlink in to the target directory of the old symlink.
Example of atomically switching a website to a new version:
Original setup - website is stored in www1 directory, vhost pointing at www symlink:
ln -s www1 www
Browse to website, see old version.
Put new website files in new www2 directory.
Set up new symlink to new website:
ln -s www_new www2
Move www symlink to directory of new website:
mv -T www_new www
Browse to website, see new version immediately.