Tableau Desktop: All your development is done here. From creating reports, charts, formatting them, putting them together as a dashboard all the work is done on Tableau Desktop.
Tableau Public:- It's a free product from Tableau. Visualization Enthusiast can play with data using Tableau Public. It has its limitation. You can only use limited data(no. of rows), can only connect excel or text file (no database connectivity) & anything you save in Tableau Public will be saved on Tableau Public Sever which anyone can download (no confidentiality).