I bet you have an Nvidia card. It's probably a GTX 670, or it's overclocked. There are numerous reports of this happening, if you look at the system log you'll see errors where the graphics driver recovered from a serious error. What's happening is, the GPU is crashing, the driver resets, and the game keeps on going but after the reset the GPU is set to the low-power 3D clock to prevent another crash, which is why your framerate is now a smoking crater. Also, if you're like me and use your HDMI output and your sound routes through the graphics card, suddenly sound isn't working anymore.
If you are OCed, you'll need to back the OC down a little. My GTX 680 was giving me the same problem as you until I backed my GPU core OC down a notch, then the problem went away. Some people with stock-clocked GTX 670s are also reporting this problem, and lowering the clock below stock resolves it. I'm not sure if this is a real problem with GTX 670s having their stock clocks set too high for what the cores can handle, or something isolated to FFXIV.