What is the wisdom behind some prayers being audible and others silent?

The juristic principle (wisdom) among the Usuliyyin (scholars of legal theory) is the benefit that the Lawgiver intended to achieve or the harm that He intended to repel by legislating a ruling. The فقهاء (jurists) differed on whether any شرعي (legal) ruling is devoid of wisdom. Ibn al-Qayyim and Ibn Taymiyyah held the view that every legal ruling has a meaning and a wisdom. Acts of worship through specific actions, such as limiting ablution to certain limbs or prayer to a specific form, are intended by the Lawgiver because they increase faith and submission to what the intellect cannot grasp. The servant needs to comprehend the secrets of the legislation to increase his certainty, and he needs to submit to what he cannot grasp to test the sincerity of his faith. Al-Sarakhsi mentioned that the reason for the revelation of "And do not recite [too] loudly in your prayer or [too] quietly but seek between that an [intermediate] way" was the harm inflicted by the polytheists. So, the Prophet, peace be upon him, used to lower his voice in the Dhuhr and Asr prayers and raise it in the Maghrib, Fajr, and Isha prayers due to the presence of the polytheists or their preoccupation.