Tableau Desktop = create reports/graphs/etc. This gives you the full power to change everything about a report.
Tableau PUBLIC = the public server where everyone can see what you do. Odds are your company won't allow you to use this for much if anything.
Tableau Server = a private server that you use to publish reports to people you choose. You can restrict their permissions so that they cannot change reports / update data but can still filter/sort certain things they want to see. (There are more options than that - just as an example)