Republish:

As name implies, it will publish every item no matter whether it is changed or not. It is intended to be used when you are PUBLISHING a new site first time. This is most time CONSUMING publishing method as it blindly publish all iems.

Smart Publish:

This method works smartly by comparing each item in the master database with the item in web database. SITECORE maintains revision number for each item which gets changed whenever the item gets updated. By comparing this revision number with web database, it will create a list of updated items changes and will publish only those items changes. Even though this method is comparing each item, it is much faster than republish method.

Incremental Publish:

Every time an item is changed, it is added to the publishing queue. This applies both to changes made through the Sitecore USER interface and changes made programmatically. Doing an incremental publish will only publish the items in the publishing queue.

Therefore only items that has been changed will be published and Sitecore does not have to do any comparisons to figure out which items has been changed. This way of publishing is therefore by far the fastest. Republish and smart publish do not USE the publishing queue.

Republish:

As name implies, it will publish every item no matter whether it is changed or not. It is intended to be used when you are publishing a new site first time. This is most time consuming publishing method as it blindly publish all iems.

Smart Publish:

This method works smartly by comparing each item in the master database with the item in web database. Sitecore maintains revision number for each item which gets changed whenever the item gets updated. By comparing this revision number with web database, it will create a list of updated items changes and will publish only those items changes. Even though this method is comparing each item, it is much faster than republish method.

Incremental Publish:

Every time an item is changed, it is added to the publishing queue. This applies both to changes made through the Sitecore user interface and changes made programmatically. Doing an incremental publish will only publish the items in the publishing queue.

Therefore only items that has been changed will be published and Sitecore does not have to do any comparisons to figure out which items has been changed. This way of publishing is therefore by far the fastest. Republish and smart publish do not use the publishing queue.