However, there are a couple of ways that this game actually gets easier when you move up the difficulty levels.
First, the principle example: there are two fundamental concepts in the game: a.) this is a wide-strategy game, having more cities means being more powerful and winning becomes easier. b.) the AI is absolutely atrocious at combat. Putting those two things together, higher level AI get so many cities out, and they do it so quickly, that successfully running a campaign against them gets you so much power, and isn't that difficult since they are so bad at combat.
Second, wonders. As the difficulty levels progress, the AI gets these earlier and earlier. Combined with the first point, this just means that YOU get to take advantage of the wonder bonuses earlier because they are building them for you.
Third, espionage. It's kind of useless at the lower levels; you're usually just defending your own cities because by the time espionage rolls around, you're already dominant. But at the higher levels, OMG what a difference they can make. free great works, tech boosts, almost unlimited gold, it's such a potent part of the strategy at high levels and just a defensive aside at lower levels.
Fourth, admittedly I'm stretching here as it only applies to one civ, Russian trade routes. The AI starts with something like 4 extra techs and civics at deity, and because of the three city starts, they usually pull away from you in the very beginning of the game. Their advantages just add to your trade route bonus as Russia.