Why does realism matter in a game? It's just a game, and it has magic; so who cares if it is realistic?
I've come up with two main reasons: your players must be able to predict the consequences of their actions, and they must feel responsible for those consequences. Both of these are necessary to make a fun game (as opposed to a narrative, where the players are merely passively listening to a story).
Decisions (picking your own battles)
Games are about making choices. But it's only a choice if you understand the consequences. If you want to play a character that bullies the little people and knuckles under to the big shots, you have to be able to tell which are the little people and which are the big shots. If taverns are run by retired 9th level fighters, while kingdoms are run by 3rd level aristocrats, guess what: you can't tell.
D&D is a game about power; and therefore, you must pay particular attention to the power distribution in your world. City guards can't be 1st level here and 5th level in the next kingdom, nor can they auto-level as your players go up in level. The wizard with a shop in town must not be more powerful than the Duke and his entire army, and yet pay taxes to the Duke. People must act like people, and that means the powerful must be at the top.
Unfortunately, the introduction of the NPC classes convinced DMs everywhere to use them for NPCs. These classes are deliberately gimped so as to be the same level as a real class, and yet never an actual threat to a real class. That, in and of itself, is stupid: why would anyone level in Warrior when they could be a Fighter? Now, you might say they just don't train as hard or have as much native talent as the PCs - but then, why not simply make them a lower level Fighter?
So now we have entire kingdoms run by NPCs armies and nobles, threatened by both goblins and demons, and utterly unable to defend themselves. They have to depend on the low-level PCs to kill the goblin tribe, and conveniently, the demons will wait until the PCs are high level before attacking the town.
This makes no sense. And because of that, it diminishes the choices the characters make. The contract that WotC enforced on DMs is called "level-appropriate challenges," and what it means is that nothing the PCs do ever matters, because the next encounter will still be level-appropriate. They can't get in over their heads, which might be a good thing, until you realize they can't knock one out of the park, either. You can't have reward without risk. The risk in a realistic game world is that your players might do something terribly clever, and defeat the demons at 1st level (like my party once defeated a dragon at 1st level with borrowed equipment and a really good plan). This could upset your campaign plans. On the other hand, your players will love it.
The other risk is that your players could make a mess of it and die. If the first thing they do is threaten the king, then they are going to be hung, drawn, quartered, and burned beyond all hope of a Raise Dead. This could also upset your campaign plans, and not incidentally, your players.
But here's the secret: if they know the King is a bad-ass - if they know the guy is the King specifically because of his bad-assery - then they will blame themselves when the King chops them down for insolence.
And your players will love it.
(Tomorrow: part 2!)