Although the Mishna’s concern is whether a person is allowed to write tefillin or mezuzot on Hol HaMoed (we follow the opinion of Rabbi Yehuda who permits a sofer (scribe) to write for himself, but not for reasons of business), the issue that is raised by the rishonim is one that is not discussed by the Talmud at all. Is a person obligated in the commandment of tefillin during this holiday period?
Many of the rishonim (including the Ri”f, the Rambam, the Rashba and others) rule that Hol HaMoed is considered to be Yom Tov, and thus tefillin should not be worn. The Ritva and others rule that these are more similar to regular days and tefillin should be worn. According to the Meiri, this was, in fact, common practice in Ashkenazi communities, while the Sefardic communities followed the ruling of the Rambam and did not wear tefillin. Another suggestion that appears in the rishonim is that tefillin should be put on but no blessings should be recited, since we are not certain whether there is truly an obligation on these days.
There is no final ruling on this disagreement even today, and different communities follow their own traditions. Most Ashkenazi communities follow the Rema who rules that tefillin are put on with the normal blessings, while the Sefardic communities do not put them on at all. With regard to this discussion, Hassidim have accepted the Sefardic tradition and do not put on tefillin. Although in the Diaspora you will often find a single synagogue where some congregants will be praying in tefillin on Hol HaMoed and others will be praying without them, in Israel the accepted practice is that no one wears them publicly.
According to those who rule that tefillin are worn, the discussion in the Mishna makes perfect sense – a sofer may need to write tefillin for himself, since he needs to use them on Hol HaMoed. Those who believe that tefillin are not a mitzva on Hol HaMoed will be forced to explain that we permit the sofer to write tefillin for use immediately after the holiday.