Campaign diff really means "everyone gangs up on you", without the AI getting any advantage.

Battle AI means relatively smarter strategical AI (operative word being "relatively" ), more stuff for you to deal with (fatigue and morale) and DEADLY tactical AI. Sending spearmen against spearmen is likely to net a loss for you, but not because the AI ones have better stats, only because the AI can micro its units (all of them, all the time too) way better that you can, taking advantage of every single piece of statistical advantage it can get from terrain, weather, precise positionning, it knows when and how to withdraw to recharge, at what range to charge etc... whereas you and I have to rely on fuzzy knowledge and rough guidelines.

It's not cheating per se, and if you have an obvious strategical advantage you ARE going to win a fight between two statistically equivalent units, but all other things being equal the AI's probably gonna get a narrow victory out of a 50/50 on paper, out of micro alone.