Save allows you to update a survey or save a copy of it. When you click Save, you will have the option to save (locking in changes to the existing survey) or save a copy, creating a new survey based on the original which can then be sent out or edited.
Publish allows you to use the same survey across EduSourced workspaces. Note that the Publish button only appears for you if your account is linked to multiple workspaces as an Admin. Publishing is useful for issuing a consistent survey across courses or programs. A good example of this would be a client satisfaction survey that you want all project clients to take, regardless of which experiential program they are working with.
Note: neither saving nor publishing affects surveys that have already been sent. These features update or create new survey templates which can be sent in the future. Save and Publish are available when editing a survey via buttons at the bottom of page.