The author function, the_author(), is used to echo the author and not return it. This is from the old functionality predating get_the_author() but is designed to echo the same value for backwards compatibility. The get_the_author() function will retrieve the author of the current post.
This function is used to echo an HTML link to the author's homepage if they have one, if not it will echo their name. The echo for this function is get_the_author_link.
The function the_author() will only retrieve the author's name, while the_author_link will create a link to the homepage of the author when available and act as the_author if one isn't available.
These functions allow developers to display the author's name and homepage on their posts.