It affects damage absorbed by enemies and their total health for certain. There is one battle in front of a cave where I just couldn't beat such a large number of enemies on Normal, had to swap to easy for that one encounter. Combat is still difficult on easy, and still forces you to fight enemies your level. Anything more than a level higher isn't happening.
I am fighting on normal constantly enemies up to 4 level above me, 3 level above me even in higher numbers. It works fine if you have enough board control to deal with them one after another. Wet + Chill = Frozen, Wet + Lighting = Stunned, Rain makes board control so easy. Add a decent initiative and 2 glass cannons mages and dealing with large groups of even higher level enemies becomes quite manageable. But your damage output suffers a lot and if your coop partner is sleeping with his CC it can become quite deadly very fast.
Anyway, the main difference seems to be your own HP. You have a 15% HP bonus on normal, and -25% HP on Hard for example. That is additive not multiplicative, so glass cannon + hard means -75% HP for example. Glass Cannon + Hard + Lonewolf means just 95% HP, add picture of health and you are at 120%, etc