I'm really confused as to why the ATI Control Center would interfere with the game downloading. Have you performed a virus scan lately? Run Malwarebytes Anti-Malware? Spybot Search & Destroy? C-Cleaner? I recommend starting by running those programs, they're free to download and they're pretty good.
If you're able to play just fine for a while and then it freezes I'd start monitoring your temperatures. Look up the program GPU-Z and check the box for logging to a file while you run the game. When it freezes, reboot and load up that log file. Your GPU should run ~60° C to ~80° C. If it's running over 100° C then check the fans on your card to make sure they're running and not blocked by dust.
It might not hurt to update your drivers either. Go to ATI's website (it'll be AMD's site) and they should have a utility that'll scan to detect your card and check to see if there's a driver update, then download and install it. This will also install the ATI Control Center so be aware of that. Your drivers should work and your computer should not lock up with the ATI Control Center running...there's no incompatibility between FFXIV and that utility so the conflict lies elsewhere.