Why is it disliked to raise one's voice in supplication (Du'a), while imams raise their voices with it in Qunut and Tarawih prayers?

The words you mentioned in your question were cited by Ibn Rajab in Fath al-Bari as a commentary on the narration of Ibn Abbas, may Allah be pleased with them both, which indicates raising the voice with the takbir after the obligatory prayer.

Ibn Rajab also mentioned the difference of opinion among scholars regarding the ruling on vocalizing dhikr after prayer, and that it is preferable to do so silently, citing Allah's saying: "And remember your Lord within yourself in humility and in fear" (Al-A'raf: 205), and "Call upon your Lord in humility and secretly" (Al-A'raf: 55), and the Prophet's (peace be upon him) saying to one who vocalized dhikr: "Indeed, you are not calling upon one who is deaf or absent."

Qadi Abu Ya'la mentioned that the apparent meaning of Ahmad's words is that it is sunnah for the imam to vocalize dhikr and supplication after prayers in a way that the congregants hear, and he mentioned texts from Ahmad indicating that he would vocalize some dhikr and secretly supplicate, and that this is not exclusive to the imam.

As for supplication, the Sunnah is to keep it hidden, citing Allah's saying: "And do not recite [too loudly] in your prayer or [too softly]" (Al-Isra: 110).

This has been challenged in light of vocalizing the Qunut, but the correct view is that the Qunut is outside of this context, because its vocalization has been proven by specific evidence. Al-Nawawi mentioned that the hadith of the Prophet's Qunut when the reciters were killed implies that he used to vocalize it, and that the correct view is the desirability of vocalizing it. In Al-Bukhari, the Prophet (peace be upon him) vocalized the Qunut in Qunut al-Nazila.

Ibn Qudama mentioned that if the imam begins the Qunut, those behind him should say "Ameen," and that there is no disagreement on this, and this indicates that the imam vocalizes the Qunut and the congregants hear his supplication, and this has been the practice of many imams of the Muslims.