Why must a veiled woman uncover her face and hands during prayer and Hajj?

A woman is not commanded to uncover her face and hands in prayer and Hajj. The majority of scholars have held that they are not 'awrah (parts of the body that must be covered) in prayer, and this was preferred by Sheikh al-Islam Ibn Taymiyyah. In Hajj, she is prohibited from wearing the niqab, burqa, and gloves. However, she is not prohibited from covering her hands when it is cold, nor from covering her face when mixing with non-mahram men by letting her jilbab fall over her face, as Aisha (may Allah be pleased with her) did.