Software firewalls are known to block executables when they detect they've changed (as in after an update). IP's can get erroneously blocked by managed lists for various other security devices/apps as well (not just firewalls, but AV, anti-malware, peer-blocking software, etc.). AV software has at times suddenly trapped execuables it used to allow after an update because of false-positive detections. So, it could have very well been suddenly blocked out of the blue by your security environment, depending on all elements in play. If you've disabled all such security, that narrows the field a good bit, but there are other things that could also be at fault.
Bad DNS cache entry (clear it), bad DNS for that matter since the app polls the FQDN and not a hardcoded IP address--hence the earlier recommendation to whitelist frontier.ffxiv.com in security software. Might be worth it to try pinging that name to see if it resolves properly. It should direct you to 124.150.157.190, and should respond to pings as well. If either fails, then you have something interfering at the network level that needs to be researched.
Otherwise, it could be something at the OS or browser level that has gotten out of whack. Things like flash and such may need to be updated or re-installed. .Net or ASP support may not be installed properly. Windows Update has periodically installed hotfixes for issues with browser features that have broken the launcher at one time or another--or provided a hotfix that resolved it. May want to check the optional section if you haven't yet--think there was one in there for Win8 specifically for Flash support, as well as some .Net stuff as well.
Oh, and make sure both .Net 3.5 and 4.x is installed as well (Programs and Features, Turn Windows Features on/off section). Things from 3.5 and back are getting deprecated in 4.x, so you could suddenly find yourself loosing some stability/functionality there. If they aren't both active, tick the boxes and get their updates to see if it resolves it.