Hum. I always liked the difficulty increase in tactical combat on Classic, what annoyed me to no end was the changes to Geoscape. If you can even call the NewCom Geoscape that. It's more like a time-management and base-building minigame tacked onto the tactical game.
Classic and above severely exacerbate the problems brought forth by the basic design of the game, that being that the player is forced to fail no matter what he does. Little compartmentalized continents with airspace barriers passable only by UFOs and Skyrangers, countries that each need a satellite to watch over them regardless of their size(can I please get a copy of the satellite over Russia so I can put it over the whole of Europe?), aliens that always attack in threes and a Council that can't get the hint and build two more XCom bases even after a whole year of them doing it, the list goes on, really.
I don't mind the kind of difficulty where a missed spike in activity results in a base and a swarm of Battleships over the USA by the third month when all you have is three Interceptors - you can really look back and say "if I'd only done X, I could have prevented this". Let the aliens step up their game, let them overpower the player and keep the "you cannot win this war" feeling that the people who defend the NewCom design like, but let the player fight back, dammit. The way the game is set up, you can win every battle that you are (allowed to be) faced with, and you will still lose the war because you're being boxed in for no good reason. And returning to the original matter, this is what I dislike about Classic mode in NewCom.
With your options so limited, you have to follow a pretty specific strategy to keep the world from falling apart in the first months, effectively replacing the "strategy" with a build-order. The higher difficulties more or less remove a large part of the game because its simplicity dictates a specific course of action that needs to be taken to keep playing at that difficulty, and I really don't like that.