On what situation is "What's the damage?" preferred to be used instead of "How much is it?"?

You only ever ask "What's the damage?" AFTER you've used the service or decided to buy it and they've rung it up. You can ask "How much is it?" at any time. 

You don't, for example, ask "What's the damage?" to find out how much a bottle of wine is going to cost you off a wine list at a restaurant but instead ask it after you've eaten the meal and drunk that wine and are ready to go.

If how much means the extent, then there may not be any difference.

It is usually said in a tongue-in-cheek way, as a form of jest to offset the truth of the pain that the "damage" is causing.  The idea is simply that the cost is painful, and calling it "damage" is a safe way to express the idea. The damage refers to the damage to your finances, not the damage to whatever it is that needs to be fixed.

Mostly only used with regard to luxury items rather than necessities of life.  It is seen as crass, cold-hearted, to talk about necessities as if they are simply dollar value things.  You can use the idiom in such a situation (necessities or important requirements) if you are on pretty close terms with the other person, sometimes.

In effect, when you say "what is the damage" you are saying "what is it going to cost me in real money".  And saying it that way means that it is only a matter of money rather than anything else.  If you think of things only in terms of money, it says a lot about your character.

You could use it after your car is repaired. It would be pretty rude in a doctor's or dentist's office. It's a joke, meaning "the money this is costing is going to hurt me." 

In that sense it's only used as a joke, and you would only say it if you were very friendly with whoever you were saying it to.