The PS4 version runs very well; it's leaps and bounds over the PS3 version. It runs similar to an upper-medium-end PC but at a fraction of the cost.
It's not to say there aren't a few issues with it, but as a beta client, you'll have to expect a few rough edges. It has crashed a few times, and the touchpad can be disruptively awkward. Even if you use the pad accidentally, it sometimes blocks further controller commands, which is really frustrating. I'm also not sure how you're supposed to "submenu click" with it (left corner tap?), or drag-and-drop (tap and hold?), although it's not to say the functionality isn't there, it's just I haven't figured it out.
There also still seems to be some limits to the windows you can have open. On the PC, I normally like to have my inventory open along side my Market Board browsing so I can see all the things I have that I'd like to sell so I can note all the prices at once, but I either couldn't figure out how to do this on the PS4, or it just isn't possible. But with more space and resources, it seems like a silly "PS3 limitation" to maintain in this version.
But other than a few snafus, it performs very well. You can join raids, "fustercluck" FATEs, or hang out in high populated areas like Mor Dhona without any noticeable slowdowns. There probably are some limits to that, but it would have to be worse than anything I've seen this weekend, and it's been pretty typical. After tanking a few DFs, I mentioned I was beta testing the PS4 version, and didn't get any remarks of any unusual quirks they might have seen. Seeing the game in 1080p is a lot easier on the visual real estate than the 720p the PS3 was limited to, although the quest list is by default still pretty imposing on the right hand side. The game world itself contains a lot more detail than its predecessor, as well, making it come alive on your TV.
I almost feel like they should let current customers continue playing on the PS4 client after the beta is over, because I see no reason it should be mothballed. But if you're going to upgrade from PS3 to PS4, you'll get the client free, and have Early Access next weekend.