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Tableau Public desktop can be used to author visualisations that can be published to Tableau Public ONLY. You cannot save files locally and you cannot publish to an internal / corporate Tableau Server. All visualisations and data published to Tableau Public is visible to everyone, you cannot restrict permissions to your work.
Tableau Public desktop can connect to local, file based data sources such as Excel and CSV files. Tableau Desktop Professional allows you to also connect to data stored on servers, such as SQL Server, Oracle, and data in the cloud, such as Google Analytics and Salesforce.com
With Tableau Public desktop you are limited to a maximum of 100,000 rows of data. In Tableau Desktop there is no such limitation