I had an issue like that when version 2.5 launched.
Apparently my anti-virus/firewall suddenly decided that the FFXIV-updater was a virus, or that it was performing an "illegal action" and blocked it from connecting to the update servers.

It was a relatively simple fix:
- Shut down (or Shut off) your anti-virus/firewall for the duration of the update then turn it back on later.
- FFXIV should update normally once your anti-virus/firewall program is out of the way.

You will not need to shut it off again when you try to play FFXIV again.