They are synonyms. But since you asked about usage, I will say: 1. I have never heard anyone say "pertinacious." It is much more likely to be found in a literary work. And I would estimate that 95% of English speakers do not know what "pertinacious" means. 2. "Tenacious" and "persistent" are everyday words and may be used interchangeably. "Persistent" may be a bit more common. 3. "Tenacious" is often used to describe an animal. That is a tenacious dog. 4. By contrast, "persistent" is often used to describe a human being. Sally is a persistent person. Nevertheless, either "tenacious" or "persistent" can be used when describing a human being or an animal.