You are correct in saying that they argue, as the Gemara in fact answers. However, the Gemara at fist understood that they both say the same thing. Both Rebbi Elazar Ben Azariyah and Rebbi Yehudah say the Shlia'ach Tzibur is Motsi the individual his Chiyuv of Tefilas Musaf. Therefore there is indeed no disagreement with regards to the Halachah of Musaf b'Tsibur, which would seem to be the point under discussion in the Mishnah.

The Gemara concludes, however, that Rebbi Elazar Ben Azariyah and Rebbi Yehudah argue about the Chiyuv Musaf of an individual where no Tsibur is present. Rebbi Yehudah says he is Chayav and Rebbi Elazar Ben Azariyah says he is Patur.

Dov Freedman