the_author() Function
The the_author() function in WordPress will simply show the display name of the author of each post. This value can either be echoed or returned as a value.
the_author_link() Function
the_author_link() can display either the author's link or the author's name. If the author has a home page set up, it will show an HTML link, otherwise it will display the author's name.
Compare the Two Functions
Both the_author() and the_author_link() provide the display name of the author of the current post. Using the_author() allows you to only return the value instead of echoing it, whereas the other function only echoes a link or name. Additionally, the_author_link() is useful when we have defined home pages for each author as it will automatically add a link for that. the_author() does not have this functionality.
Summary
Both of these functions are useful for designing how posts show up in WordPress. It is important to understand the different uses for each of them to get the right functionality for a website.