I'm a couple of hundred turns into my first play through on prince. I'm absolutely loving it so far, though it absolutely needs a patch.
The AI is very irrational and thick. The Germans declared war on me from the other side of the map, so I ignored them as it would take an age to get to me. After 40 turns they arrived at my borders with an army big enough to steam-roller the entire map. I moved my only two units (archer and knight) within range and took out two of their units... they immediately sued for peace! Great for me, but didn't make sense.
I was also allied with France, who I couldn't possibly have a better relationship with, and they declared war for no reason at all. We shared borders with each other, but they never sent a unit to attack. I can only assume the declaration, and then lack of action was a bug rather than intentional mechanism.
I absolutely love the new districts too - makes planning cities for more logical as I can easily see what districts a city has, and therefore what my intentions were for it. In Civ 5 my plans would always go out of the window late game, as I just started to get bored of remembering what I wanted a city to focus on, and built everything possible.
The other change I think is top notch is the builders. At first, I thought it would be a terrible change, but with the right civics, being able to build 5 improvements over five turns takes away some of that tedium.
There's a few other bugs I have in my game, but not enough to say the game is broken. I already think it is on par with Civ 4 & 5, which for a Civ game on release means in a years time, it will be godly.