Explain the_author() Function
This function, the_author(), displays the name of the author on the current WordPress post. This function is based off of an older function called get_the_author which may be used in older WordPress themes. The two functions are very similar and have the same purpose.
Explain the_author_link() Function
This function, the_author_link(), displays the name of the author which includes an HTML link to the author's home page if they have one. If there is not an author's home page set up, then it will just display the author's name with no link built in.
Differences/Similarites
The two functions are similar because they will both at least display the author's name. However, the main difference is that one includes an HTML link if the author's home page is set up.
Summary
While these two functions are very similar, I think I would only use the function, the_author_link(), if the website was being used for blogging purposes. Many people may want to know more about the author if they come to the website just to read the blog. Otherwise I think that the function, the_author, will serve the purpose most websites need - to display the author's name.