The tool you are describing is "rsync".
It's something which started life on Unix but as with all useful tools has been ported to most operating systems now, including Windows.
Rsync offers a wide variety of options, including to remove files from the destination when they are no longer on the source, as in your example.
As for which version of rsync to use, I'm afraid I really can't advise, each has different strengths and weaknesses and I'd recommend you try them to see which best fits your preferences.
I can say that cygwin (Virtually all Unix tools ported to Windows) is very good, and worth a look.