אסור לילך בקומה זקופה, ולא ילך ארבע אמות בגילוי הראש
2:6 "It is forbidden to walk in an upright posture, and a man should not walk with his head uncovered for four cubits (6 feet)."
The textual source of these two Halakhot is the same: TB, Qiddushin 31a. The trained student, however, will notice a difference in the way these two rules are presented. While walking in an upright position is classified as "forbidden", walking with the head uncovered is not.
All Rabbinical laws have their origin in the Talmud, and the later Rabbis paid close attention to the way the laws were formulated. In the case of walking upright the Talmud quotes rabbi Yehoshua ben Levy who said: "It is forbidden to walk erected". Rashi explains that in light of God's Omnipresence walking in an upright position is considered a sign of arrogance. We should walk with humbleness.
On the other hand, when talking about covering one's head the Talmud does not bring a general rule, but the case of Rab Huna. "Rab Huna never walked [even] four cubits without covering his head. Rab Huna said: God's Presence is above me". In another text Rab Huna emphasized the exceptional nature of his behavior: "I should be rewarded for never have walked four cubits with my head uncovered".
All rabbis agreed that as opposed to "walking upright", "walking with the head uncovered" is not a universal prohibition but Middat Hasidut: a personal stringency. Maimonides classified this kind of personal regulations, brought very often in the Talmud, as mandatory for Tora Scholars. Maimonides dedicated one whole Chapter, Hilkhot De'ot Chapter 5, to the standards of behavior of Talmide Hakhamim (Tora Scholars) which are obviously, way above the normal standards. In Halakha 6 he mentions that a Tora Scholar should not uncover his head.
On the other hand, and formulated as "one should not", the Shulhan 'arukh, following the Tur, did no limit this indication to Talmide Hakhamim but to everyone else, as a permanent reminder of God's presence.
NOTE: The above mentioned debate refers to "walking" with the head uncovered (at the workplace, in the street, etc.) . However, in the Synagogue, when praying or when studying Tora, the head must be covered.