Tableau Public
This is essentially a free version of Tableau visualization software. It allows you to use the majority of the software functions.You can create visualizations and connect to CSVs, Text and Excel documents.

However, the largest difference is that Tableau Public does not allow you to save your workbooks locally. You have to save them on the publicly which means that everyone can see your data since it’s saved on the cloud.
Tableau Desktop
Tableau desktop offers all the full features of software. Your workbooks can be share locally. This version allows you the connect to different file types, create extracts of the data sources and save your Tableau workbooks locally and publicly.
The only thing that is missing here is the data connection types. You will not be able to have a direct connection to a database or web and software APIs.
Tableau Reader
Tableau Reader allows you to read the Tableau file types. If you want to share your workbook by sending a file, the receiver will need a Tableau reader to open the document. Without the reader, you may need share it publicly or convert the workbook into a PDF format. Learn more info from Tableau Training
Tableau Server
Tableau Server allows users to save workbooks securely across the organization using a secure server. This eliminates the need for a user to have to share the workbook publicly. However, this comes at additional cost.
To get in-depth knowledge, enroll for a live free demo on Tableau online Training