The Save widget allows logged in users to save the following in a map with additional graphics, different layers or a different extent. This :
Additional graphics added via the Draw/Measure widget
Layers turned on or off in the underlying webmap i.e by using the layers widget
A different map extent.
The following are not saved when creating a Saved Map
A different basemap
Changing Layer transparency
Changing Layer visibility
Additional layers added via the add data widget
A saved map can be kept private for use again when logged into LocalMaps or can be shared with another user - whilst maintaining the additional information.
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