rsync does all that.
I backup my mainfolder (and subfolders ) to my NAS1 and NAS2. For NAS1 I have the --delete flag so mainfolder and NAS1 are always the same. diff mainfolder NAS1 shows me all dirs/files are common.
But for NAS2 I don't have the --delete flag so NAS2 will contain other files, folders and eventually grows. If there is a file with same name, rsync will replace on DEST. diff mainfolder NAS2 will show the common and differences.
I suggest make two folders, /home/smith/mainfolder and /home/smith/nas1, throw some files in mainfolder, rsync mainfolder nas1. Modify files in mainfolder, rsync again, see the result.
Do, diff mainfolder nas1
The Web has plenty of examples how to use rsync.