This function is used so that you can control what the authors name is displayed on each post.
When you use this function it will create a link from clicking the authors name on the post. If the author has a home page then you can link to their home page through the link or it will just return the authors name.
The differences are the_author() function will help you control how you want to show your name on posts determine how others see it. Then the_author_link() will search to see if the author has a url for their profile page and then it will use that to link to their page, but if there are no url's for the author then it will refer to the_author() function to just provide the name.
These two functions are very useful because you want to make sure that the right things appear on the post. You wouldn't want your username to appear for everyone to see, so using the right display name through the function is important. Then same goes with the link function since you want to lead to the right page and show the correct display name based off of the_author() function.