I don’t use the terms synonomously. It is true that people who are puritanical are possibly very likely to follow rules of modesty, but modest people are not necessarily puritanical. I use puritanical related to religion, probably only with Christianity, but maybe I am wrong to do that? I would never call a Chassidic woman puritanical, even though she dresses extremely modestly, has specific beliefs about God, sex, and nudity. I guess I agree with some of the answers above, it has to do with the person wanting others to behave the same way.
Time and place issues are exercised even by people who might dress fairly modestly at times. They might be topless at the beach, but never show cleavage in daily life. I don’t really think of that person as modest, but rather that she dresses modestly in certain situations.
A person who always dresses for modesty reasons, for religious reasons, I would probably clarify it is for religious reasons.
I also use puritanical to describe my disgust about not being able to discuss health issues regarding sex and women’s girly parts. Of course this is changing in America, but still sexual health discussion is a no no in many circles. When AIDS hit the scene, Reagan and his administration were not going to go on TV and talk about that gay disease and condoms, no matter how much our Surgeon General thought it was important to inform the public. Thank goodness the Hollywood community spoke out, produced films, and were very vocal. Still now there is a strong resistance to teach sexual health to our children.