I had the same problem and solved it easy.

Go to C:\Users\YOUR NAME HERE\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn and rename the screenshots Folder into Screenshots.
Then open ingame the System Configuration / Other Settings / Screenshot Folder.
Click it and delete the Location from your Screenshot Folder and replace it with C:\Users\YOUR NAME HERE\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\Screenshots.
Click apply and thank me later.

Your Name here is your Computer Name.