Rabbi Dovid Rosman (who authored an essay on Migdol vs. Magdil that this article is based on) thinks that we say Magdil during the week because Dovid said this version of the shirah before he became king, while the version in Sefer Shmuel was said after. The Avudraham explains that Magdil indicates a king who has a small kingdom while Migdol indicates a king with a large kingdom. Shabbos is compared to a great king, so we say Migdol on Shabbos.
Finally, the Iyun Tefilah says that Hashem is creating the geulah while we are in golus; He is setting the stage and building our salvation – He is being magdil the yeshuah. When the geulah will come it will be like a migdol, a strong tower. On Shabbos, we are living in a time of mini-geulah so we say Migdol.
When we say Migdol in bentching this week, we will have a much greater understanding as to why.