When switching from any difficulty to peaceful in the nether or end dimension, all mobs despawn as expected. Switching to easy afterwards does not spawn mobs again though (switching to normal or hard does). As far as it seems to me, the issue can only be solved by exiting and reentering the world (no full game restart needed). The tested mobs where Zombie Pigmen and Endermen, but the issue probably also affects others.

To clearify the issue, when switching difficulty in the order anything but peaceful to peaceful to easy (and always continuing the game inbetween to allow for (de-)spawning), the behaviour is as follows:
anything but peaceful to peaceful: mobs despawn as expected
peaceful to easy: no new mobs spawn (even though they should)

When switching difficulty in the order anything but peaceful to peaceful to normal or hard to easy (and always continuing the game inbetween to allow for (de-)spawning), the behaviour is as follows:
anything but peaceful to peaceful: mobs despawn as expected
peaceful to normal or hard: mobs spawn again as expected
normal or hard to easy: spawned mobs do not directly despawn (only when leaving the area, as expected), but no new mobs spawn (even though they should)

Even though the issue does not seem to affect the overworld, leaving and reentering the nether while not on peaceful difficulty does not seem to fix the issue. (i.e. nether peaceful to nether easy to overworld easy to nether easy). Even though not tested, the same behaviour probably does not fix the same issue in the end aswell.