What are the differences between Minecraft difficulties?

Normal

Hostile mobs spawn and deal standard damage. The hunger bar can deplete and will leave the player with 1 half heart if it drains completely. Villagers killed by zombies have a 50% chance of turning into a zombie villager.

Hard

Hostile mobs spawn and deal greater damage than on Normal difficulty. The hunger bar can deplete and the player will starve to death if it drains completely. Zombies can break through wooden doors and can spawn reinforcements when attacked. Villagers killed by zombies have a 100% chance of turning into a zombie villager. Spiders can spawn with status effects.

Copied off the Minecraft wiki.

As you can read from Minecraft's wiki, the main differences are:

• Hunger can kill you
• Zombies are tougher (break through doors and spawn reinforcements)
• Villagers always turn into zombies when attacked by zombies
• Spiders have status effects.

There are quite a few differences between difficulty modes. I'll try to mention all the major ones here. Information was found from an article on the Minecraft Wiki.

Peaceful

This is the easiest mode. Monsters do not spawn, the health bar does not deplete, and health is regained very quickly. No mobs deal any damage at all.

Easy

In easy mode, mobs do spawn but don't deal as much damage as normal or hard. Cave spiders can't poison you, withers do not give the wither effect, and zombies don't turn villagers into zombie villagers. Your health bar does run out, but you will not receive damage from starvation if you have less than 10 health points (5 hearts).

Normal

Mobs do more damage than in easy mode. Villagers turn into zombie villagers 50% of the time (otherwise they just die). Cave spiders can poison you and withers can inflict the wither effect. You can be brought down to one health point (half a heart) from starvation.

Hard

Mobs deal the most damage. Zombies can break through wooden doors. Villagers always turn into zombie villagers when killed by a zombie. You can die from starvation. Spiders can spawn with status effects.

Hardcore

This is actually an entire gamemode just like Survival and Creative, not a difficulty level. In Hardcore, commands are disabled, the difficulty is locked to Hard, and you have only one life. Once you die you can either change the gamemode to spectator or delete the world.