Ran into this same issue on my Mac. I usually run in fullscreen mode, but switched to windowed for a session, when I tried to start the client again it would run (music and all), but I couldn't see the window unless I used Mission Control to switch between desktops.

I fixed this by looking in the file

/Users/.../Library/Application Support/FINAL FANTASY XIV/p_drive/User/My Documents/My Games/FINAL FANTASY XIV - A Realm Reborn/FFXIV.cfg

Here "ScreenLeft" was set to some enormous number, which I'm guessing explains why I couldn't see the window anywhere. I just edited the file to set "ScreenMode" to 1 instead of 0, and the game launched successfully in fullscreen. Fixing "ScreenLeft" and starting in windowed mode might work too, but I didn't try it.