Who would wear Rabbenu Tam tefillin and why?

The Shulchan Aruch (Orach Chayyom 34 3) writes that only someone well-known for his outstanding piety should wear tefillin of Rabbenu Tam. The halacha is that the mitzva of tefillin requires wearing tefillin only as described by Rashi and Rambam. Since the halacha is not like Rabbenu Tam, the question is why is there a custom to wear Tefillin of Rabbenu Tam at all? HaRav Moshe Feinstein (Iggrot Moshe Orach Chayyim IV 9) explains that the reason for this is that there were many communities at the time of the rishonim and before whose psak halacha was like that of Rabbenu Tam on this matter. There are explanations for two sets of tefillin based on kabbalah and there are hassidic customs where the wearing of tefillin of Rabbenu Tam is more widespread (see Shaarei Teshuva 34 6).