Just to rule it out, make sure you always launch as an admin (right-click and run as admin, or open the properties page and set it to always run as admin). Some files are in protected areas that may need admin rights to write data to them. Also make sure that all the executables in the installation folder are allowed network access in all your security--that may be preventing a proper update as well. Don't forget to check Windows Defender--many don't even realize this is installed and running on their systems by default (I just disable it myself, since I use third party security software).

Double check the version strings in the current and backup info files. You can open them in any text editor... word, notepad, wordpad, textpad, etc. Each .ver/.bck set should have the same version string. If not, make note of them and then set them both to the older/lower string and save them. Try again while launching as Admin and if it still errors out, try setting them to the newer/higher date/version string and try again. It may cause you to go through and patch again, but it may be better than a full re-install.

Program Files (x86)\SquareEnix\FINAL FANTASY XIV - A Realm Reborn\game\ffxivgame.ver and ffxivgame.bck

Program Files (x86)\SquareEnix\FINAL FANTASY XIV - A Realm Reborn\boot\ffxivboot.ver and ffxivboot.bck