They are synonyms and can be used interchangeably. See http://www.collinsdictionary.com/dictionary/english/stubborn

here in Brazil we use stubborn when somebody do something wrong and continues doing it even you say what is the right way, exemple, you say me that 2+2 is 3 and I correct you and show that is 4, but u don't continues saying that is 3, so you are stubborn at least we use like that in Brazil ahahahah And Obstinate is who persists, example, you want to be a well-known person, so you study a lot and get good scores, make good projects and stands out from other, you are obstinate ahahaha well I hope helped you o/