Although the Gemara (150a) rules that it is permitted to think of weekday matters on Shabbos, it subsequently (150b) cites the episode of a certain very pious individual who found himself thinking of his weekday matters on Shabbos and felt that he had violated the sanctity of Shabbos by doing so.
Reb Tzadok HaKohen of Lublin (Yisrael Kedoshim #8) writes that when we thank Hashem in the Minchah of Shabbos for providing us with the potential to achieve a מנוחת אמת ואמונה — a rest of truth and belief — we are referring to our potential to attain the level of that very pious individual. For the ideal Menuchas Shabbos in which even in the depth of our hearts we regard all weekday matters as having been completed and no longer requiring our attention. It does not suffice for us to do the minimum, to refrain from forbidden activities yet at the same time allow our hearts to continue being distracted by our commerce and craft.
Only such a state can rightly be termed a Menuchas Emes, for truth is that which penetrates to our greatest depths and remains consistent. A “truth” that only “holds true” on a superificial level (refraining from expressing concern over weekday matters, yet at the same time considering, these matters in our mind) cannot be called “true.” Such superficial truth might fool other people, but not Hashem, concerning whom it is said שחותמו אמת — his seal is truth.
Thus, while refraining from mundane conversation is a technical fulfillment of the command to desist from expressing oneself in weekday matters, it is not a Menuchah Sheleimah (a truly complete, befiting rest).
Hence, concludes Reb Tzadok, those who are in awe of G-d because they are cognizant of Hashem’s knowledge of that which is concealed deep in one’s heart are extremely meticulous and vigilant in barring any weekday thoughts from their minds.