On Battlefield:
Easy: AI gets -4 attack
Medium: No bonus/penalty
Hard: AI gets +4 attack
Very Hard: AI gets +7 attack
That's all, no morale or anything.
Note that the numbers are not 100% exact, as the bonus is given through a parametre in the chance_to_kill formula, which produces for example 3,899 (which roughly equals 4). If you're interested, the formula is near the bottom of the original post of this thread, under 'useful formulae'.
Campaign:
The harder the difficulty, the more aggressive the AI towards you, the more difficult is to make alliances and stop wars involving you, and (I think) it gets more money bonus per turn.
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