Both words have near exact meanings; however, Obstinacy is a noun and Obstinate is an adjective, so the way you use these 2 words will differ.

Example #1: Obstinate

Obstinate and unyielding, the judge refused to give the defendant credit for time served. (Obstinate is an adjective. Adjectives describe nouns. Obstinate clearly describes the judge's unwillingness to give the defendant credit for time served.)

Example #2: Obstinacy

No matter what logic or rationale I used, nothing I came up with could break through her obstinacy. (You can see how obstinacy isn't an adjective. Simply replace Obstinate in the first example with Obstinacy and notice how off the sentence reads.

Obstinacy is a quality or trait, and we know a noun is a person, place, or thing. Well, a quality is a thing, making Obstinacy a noun.

P.S. I didn't write the sentences myself. I got them from this site. http://wordsinasentence.com/vocabulary-word-list/ However, I did write the explanations. :)