I've been having the same problems even since beta phase! When I try to request help, they just assume it's my internet connection even though my internet is fine when the lag occurs and alt-tabbing has never fixed slow internet before. :P
I'm using an AMD Radeon HD 7000 series card though so I'm not sure what the problem is. I originally assumed that my card just stinks, but it's nice to know I'm not the only person having this problem. Even if I alt-tab to a smaller window (so that I can still see the FFXIV screen), I can see the lag go away and the game catch up super fast to what it was doing. But as soon as I alt-tab back into the game, the massive lag is back. If I wait too long without alt-tabbing, it can even disconnect me. It's really weird.
And my computer was just updated a few months ago and is pretty bamf so it definitely has the power to handle FFXIV in any mode.