As piety and pity are etymologically the same word (see my post on this pair), so providence and prudence are etymologically identical, both deriving from provideo, providere (to see ahead). Prudence is etymological is simply a contraction of providence. Providence is often capitalized to render the sense of God's guiding hand or plan; prudence is used without religious connotation. This religious/secular relationship is again similar to that of piety and pity. As pity becomes pious when it is a reflection of God's grace, so prudence becomes Providential when man's practical wisdom takes into account divine purpose.
‘Pity’ and ‘piety’ both ultimately derive from the Latin pietas which, as used by Vergil, means the fulfillment of duties to family, country and gods. Aeneas is a man marked out for his pietas (pietate insignis). In Latin, pietas does occasionally suggest ‘affection’ and even ‘compassion’, especially with respect to family members and friends (vide pietas, Lewis & Short). Generally, commentators have not recognized this last occasional use,even though affection of a father for his child or the child for his father iscompletely natural within the context of familial obligation. Later, Christian writers extended the meaning of pietas to include God’s mercy. This too, however, has its parallel in classical civilization where a man’s pietas is expected to earn him a gracious response from the gods. Indeed, Aeneas complains loudly about the apparent lack of divine reciprocity for his pietas. Today, ‘piety’ is almost exclusively used within a religious context and even sometimes pejoratively connotes ‘holier-than-thou’ pretensions.
In OF there were two spellings for pietas: piete and pite, both with the sense of compassion included at first within the idea of fulfilling one’s religious duty and later simply as compassion itself. Later, ‘piete’ and ‘pite’ became differentiated, with the latter (pite/pity) becoming a secular expression.