How do different difficulties work on story missions?

First off, the enemies rank that are encountered in the mission (not in the beginning of the map) are at the same level as the difficulty

Second, sometimes they offer more rewards. Weekly Heroic at lvl 26 would give you 6 strange coins, as oppose to at lvl 28 would give you 9 strange coins. Same thing can apply to others. Daily Heroic at lvl 26 would give you 1 ascendent material, as oppose to at lvl 28 would give you 2 ascendent energy. However, many of the standard missions that can have a heroic modifier but aren't weekly or daily, can be just vanguard bounty, and give you no more reward than when run at lower difficulty.

Third, Mission Modifiers are applied to make it even harder. These Mission modifiers can be:

1. Angry, Minions of the darkness display heightened agression
2. Arc Burn, Arc Damage from any source is greatly increased
3. Epic, Heavily shielded and highly aggressive enemies appear in great numbers
4. Heroic, Enemeis appear in greater numbers and are more aggressive
5. Juggler, No ammo dropped for your equipped weapon
6. Lightswitch, Minions of the darkness deal much more melee damage
7. Nightfall, If all players die, you will be returned to orbit (<--- rough one!!)
8. Solar Burn, Solar Damage from any source is greatly increased
9. Void Burn, Void Damage from any source is greatly increased

Sorry this is in answer form and not a comment to @Eben's answer, I dont have enough rep to add a comment. But to add to what he said you also get bonus xp for completing the mission if you do it on the harder difficulty.