Rsync only sync destination folder existing files

I would use rsync as it means that if it is interrupted for any reason, then you can restart it easily with very little cost. And being rsync, it can even restart part way through a large file. As others mention, it can exclude files easily. The simplest way to preserve most things is to use the -a flag – ‘archive.’ So:

rsync -a source dest

Although UID/GID and symlinks are preserved by -a (see -lpgo), your question implies you might want a full copy of the filesystem information; and -a doesn't include hard-links, extended attributes, or ACLs (on Linux) or the above nor resource forks (on OS X.) Thus, for a robust copy of a filesystem, you'll need to include those flags:

rsync -aHAX source dest # Linux
rsync -aHE source dest  # OS X

The default cp will start again, though the -u flag will "copy only when the SOURCE file is newer than the destination file or when the destination file is missing". And the -a (archive) flag will be recursive, not recopy files if you have to restart and preserve permissions. So:

cp -au source dest

You could use -i --dry-run to get the list of changes between your two folders and then only keep the files that have been deleted.

However, most people do backups with rsync by first doing rsync and then cp -al of the backed up data (e.g. current) into a different directory (e.g. archive). The -l argument tells cp to not copy the data but to just create a hardlink. That way, you have the same data available under current and archive, but every file occupies the disk space only once (e.g. without the overhead you are experiencing now).

Later, as soon as you again rsync your server to current, rsync will update the all the files that have changed and delete the files that have been deleted. The files in archive, however, will not be touched.

Of course, you can create as many archive folders as you want. Also note that rsync offers the --link-dest option, which can automatically create the archive folder during the rsync run, without having to run cp -al. See the man page for more information.