The difficulty selection determines a few things on the map:
Starting resources-At the higher difficulty ratings you start with a little less mana, a little less gold, AND fewer "bonus" troops.
Enemy/Ally troop numbers-Your enemies have better troops and your allies worse as the difficulty rises.
Carryover-When you finish a map, I'm sure you've noticed that you can buy stuff to carry on to the next map? Well the cash on hand at the different levels is different. I was rather surprised to not be able to carry anything but my prized First Strike Lance with me the first time I played at Hard on the campaign. This is one of the biggest issues because carryover can have a big impact on your next board. I carried over four goblin wyverns once and had half the board conquered by turn 5.
AI tenacity and benefits-The AI is supposedly smarter at higher difficulty, but I would just say that it's more aggressive and tenacious. Until you knock him out of the game by winning a few key battles, he'll keep coming and coming sending flyers behind your lines to snipe resources and sending as many powerstacks as he can against you. The endgame is all the same once you get him down to a very small territory though. Instead of sacrificing L1 and L2 units to build more L3 and L4 units and summoning powerful creatures instead of junking frogs, eagles, and boars he'll just sit there with whatever he had when you put him on a 0 or negative gold/mana production rate. Since he gets more xp and more resources per turn though at higher difficulties (He'll level two to three times faster than your hero at emperor most likely), then he can have a bigger "final stack" for you to beat even if they're mostly L1 and L2 units (with a penchant for catapults over ballistas sometimes).
Overall, the carryover is the biggest hurdle to get past IMO. Really though it just makes you choose items/units that will help you the most from a smaller pool. I would have loved a first strike lance AND a centaur at the hard difficulty level, but only medium gives you enough gold .Peace...
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Feyd Rautha
"It is by caffeine alone that I set my mind in motion
It is by the beans of Java, that my thoughts acquire speed;
The hands acquire shakes; the shakes become a warning.
It is by caffeine alone that I set my mind in motion."