I had this problem Friday. Your client will still attemt to update even though the beta isnt active. Once you're patched up if you click Play it will give you an error that says "unspecified". This is because theres supposed to be an English error there but they never put it in. The error is correct if your language is set to Japanese. Its supposed to say the beta is not currently active or something.
To resolve your problem, go into your root drive\users\your user name\my documents\my games\final fanasty xiv: a realm reborn (beta)\download\ or I think its \patch\ (im not on my home computer so im going off memory here) and theres folders with what looks like random alphanumeric names and should be a file called GameInstall something. Open that GameInstall file in Notepad and look at the path of the latest entry on the list. Go rename that folder to \whatever_old\ and restart the launcher.
Also try closing everything you can before running the launcher, web browsers included. No joke.Nothing but the launcher running. Disable your anti-virus too.
Now just to note, I did that several times and got the error before it worked. It took me about 5 tries before it updated error free. I'm pretty sure that somewhere along the download gets corrupted and the error we're seeing is a checksum mismatch error. Of course since we're English we dont get a meaningful error, just cryptic numbers, so its just a guessing game as to what is actually going on.
I have a feeling a lot of English speaking people will see this upon launch and immediately be reminded of 1.0. The devs have not aknowledged this error and its not that uncommon. I have no doubt it will make it to release and they will blame the user's machine without investigating the cause of it.