Change difficulty level or mode without starting over

As with all of the Fire Emblem games, if the main Lord(s) die, there will be a game over. For Awakening, game over will only be triggered if Chrom or Avatar dies. If the other character dies, they will be gone forever in the Classic Mode (but you can still proceed with the game without them) or temporarily in the Casual Mode.

It doesn't change the story in any way but it does change the amount of enemies and the location of them on missions while making them significantly stronger and with different items + skills. The only way to get different storyline is to change your gender during another playthrough and have the characters marry different people. Please note that I haven't finished my Lunatic playthough but haven't seen any differences so far. Also, this is expected as the script writers had TONS of stuff to write already (most of which you will never be able to read), so this is most likely just a gameplay difficulty.

Spoiler: Lucina will have different dialogue with you if you are married to her throughout the game. You can also marry Chrom which will produce a slightly different story.