The Pro users are given a few more features like manual renaming, renaming depending on the post the media is attached to, logging of SQL queries and a few more options. With the Pro, a good process is to actually let the plugin do the renaming automatically (like in the free version) and to do manual renaming for the files that require fine tuning.
It automatically renames your media filenames depending on their titles. When files are renamed, the references to it are also updated (posts, pages, custom types and their metadata).
A new column in the Media Manager will display to you the new ideal filename and a button will allow you to rename it straight away. You can lock and unlock the renaming automatic process through little icons. By default, this is done automatically depending on the title of your media. If you have the Pro version, this can be done also based on the title of the post (page or any other custom type) the media is attached to.
There is also a little dashboard called File Renamer in Media that will help you rename all your files at once.