In an ideal world the jump between the difficulty levels would simply involve the computer AI being more intelligent and devious on the higher difficulty levels. Sadly and as in most games it doesn't work like that and the game simply introduces factors to balance the game against you which I think make the game a touch unrealistic. Never the less if you think it's too easy then this is perhaps what you should do.

My recommendation to anyone who finds the vanilla campaign to be too easy is to get a copy of BKBs Super Mod. It currently has versions for both Early and High periods with Late on the way. Its my favourite mod, simply for the way it introduces lots of new factions which means you constantly have to consolidate what you already have to prevent re-emergences and rebellions in your lands before you are able to move on and conquer more. It also makes using multiple points of attack tricks agains the AI less viable before borders with individual factions are often much smaller which can force you into decisions between invading multiple factions or going for a pure direct assault.

In short, you have to be more diplomatic and take care about who you kill.