According to מִלּוֹן אֶבֶן־שׁוֹשָׁן the two terms are synonyms, but have some slight differences. A פזמון is a repetitive song with a recurring chorus, usually written about religious or romantic topics, whereas זמר is a more general term used for "song."
The Jewish Encyclopedia and Wikipedia both note the distinction between Ashkenazic and Sephardic usages, but do not offer any explanation. Ashkenazim use זמר to refer to songs sung on religious occasions, like on Shabbat, and use פזמון to refer to specific liturgies in seliḥot. However, Sepharidim use the term פזמון to refer to "any important hymn, in parts of the service other than the seliḥot," much the same way that Ashkenzaim use zemirot. As Wikipedia states,
Pizmonim are traditionally associated with Middle Eastern Sephardi Jews, although they are related to Ashkenazi Jews' zemirot. The best known tradition is associated with Jews descended from Aleppo, though similar traditions exist among Iraqi Jews (where the songs are known as shbaḥoth, praises) and in North African countries.
In addition, according to Wikipedia, "Pizmonim are extra-liturgical, as distinct from piyyutim, which are hymns printed in the prayer-book and forming an integral part of the service." As such, Ashkenazim would refer to the "extra prayers on the holidays" as piyutim, not pizmonin. ("Yotz'ros" refers to the specific piyutim additions in Birchot Kriat Shema.)
This usage split dates back at least to the times of the Rishonim. According to the Jewish Encyclopedia, the Machzor Vitri, one of the earliest and most important Ashkenazic siddurim, contains zemirot, whereas a Sephardic siddur published in Venice in the 1500's makes explicit reference to pizmonim.