I saw the former for the first time today, but I noticed its definition seems exactly like the latter. Is there something I'm missing or are the two just pure synonyms?
Stubborn - A tendency of disposition to persist in an opinion, belief, decision, or course of action, generally with more force than reason.
Obstinate - A particular instance of stubborn behavior.
Dick had always been a stubborn boy, but he was particularly obstinate in his dislike of homework.
Obdurate - A hard and unyielding attitude of a harsh and unfeeling character.
The foreman was obdurate in holding to rigid production schedules.
Source: Choose the Right Word, by S.I. Hayakawa
Note: I had looked up this same cluster of synonyms in the past with Webster's dictionary of synonyms and found the same explanatory distinctions.