Just to add a little more, I think you can say “stubborn unemployment “ because “stubborn” can refer to inanimate objects, like a government (in the way you used the word), as well as people, but “obstinate” only would refer to living things, especially people. “She was just so obstinate.” That works. Or “She was just so stubborn.” You can say, “man we’ve had the most stubborn weather! It just won’t change.” Or, “that bolt is so stubborn, it won’t budge with this screwdriver.” That works, but you couldn’t say “obstinate” in those sentences.

They're interchangeable when you're talking about a person's attitude or behaviour. 'Obstinate' may be slightly stronger, but they have pretty well the same meaning. Your sentence about unemployment doesn't sound right. I have never seen 'obstinate' being used to refer to anything other than a person. 'Stubborn', on the other hand, can be used for inanimate objects. Detergent adverts often claim their products remove 'stubborn stains.'

I would say obstinante is more formal, much more elevated sounding than stubborn. With the questions about a girl being too obstinate or stubborn to change her mind they are both okay, and can be used interchangeably, but stubborn is less formal. If you’re going to be informal enough to say “bro”, which is fine, I’d definitely say you’d hear stubborn in that sentence much much more often because of the sentence’s informality. You can’t say the government had an “obstinate” problem of unemployment, but stubborn is okay. So, it’s the direct opposite of what you thought. Sorry. I don’t know exactly why, but I suspect it’s just idiomatic. It’s just like in German; some verbs match up with some words and others don’t. There’s no real explanation of why. You just have to research the word a little for examples, or ask a native speaker who can just tell you if something sounds normal or not.

They mean the same thing. I would say that obstinate is more severe than stubborn.