All comments are based on 1 incomplete King/Standard/Standard game.
Warfare. No noticeable tactical improvements so far. Unit levels seem down from prior King experience.
Expansion. Very slow to expand. I've reached modern age and I don't think an AI Civ has more than 5 cities. Its back to expansion levels prior to last years fall patch.
Use of new mechanics - I'm getting outpaced by one Civ in the tourism department, but thats about it. Granted, I rolled Germany, Carthage, Zulu, Japan, Poland, Brazil, and Inca. Inca and Brazil got beat back relatively quickly so that left the warmongers.
Diplomacy - Very friendly. 2 DOWs and I suspect more are on the way, but Civs are definitely quicker to forget. Not across the board, but can see changes in their attitude over time. Competition for city states has been intense. Haven't seen this much competition for CS favor at King.
3/4 of one game in, my overall experience is the game is easier. I've played a pretty poor game, usually if I played this poorly on King there would be at least one Civ runaway to deal with. That has not been the case so far.
I'll likely make the jump to Emperor after a couple of more rolls if only because I've more recent experience with AI performance at that level.