Game stage determines loot and enemy spawning.
Higher gamestage means higher tier loot.
Higher gamestage means stronger zombies wondering the world and spawning on blood moons.
Gamestage is calculated by your level & your days survived.
https://7daystodie.gamepedia.com/Difficulty
The wiki explains it best. However, in Alpha 19, a change not listed on the wiki. Difficulty no longer affects game stage. Meaning playing on adventurer or Insane doesn't affect gamestage, meaning it won't affect the loot or the types of wandering zombies.
Ignoring gamestage, days only affect difficulty in the following ways:
1) 1 Wandering horde per day, so each day you get 1 of them.
2) Zombie respawning as random walkers. Since the spawning is block specific, and the reset time is based on days passed. Meaning that playing with 10 minute days vs 60 minute days will spawn 6 times more zombies per realtime(However, zombies also expire, so more will die randomly to time passed)