I also started having this problem after my computer installed the latest Windows 10 update. I found that changing from DX9 mode to DX11 lets me get into the game and it seems to be working ok once I'm in.

However, when I log out of the game I get a message about an "unexpected error" occurring when exiting FFXIV. It lists a bunch of dll files as well as some ffxiv_dx11.exe files. I just click ok on those message boxes and it goes back to the desktop just fine, but it still bugs me that they are happening at all.

I've considered rolling back the Windows 10 update, but since the problems right now qualify mostly as "nuisance" I haven't bothered.