Saving in your local store means saving it in your browsers 'history'. A bit like a cookie. You will be able to access it later via the load option. IMPORTANT: deleting cookies will also delete your maps in local storage. Saving in local storage is not meant for long term, but usefull when quicksaving while working on a map.

Downloading will download the file in json format to your download folder. After downloading you can store the downloaded file anyplace you like. You will be able to load it later via the load menu and continue working with it.