Once, the Companions of the Prophet, peace be upon him, were travelling, and loudly engaging in dhikr (remembrance of Allah). The Prophet Said:
"O people! Be gentle on yourselves, for you are not calling someone who is deaf or absent. Rather, you are calling the One Who hears everything, Ever-close."
Ibn Taymiyyah delved into the wisdom of making dua silently, and mentioned a number of benefits to this:
Firstly, it is a sign of strong iman, as the person demonstrates that he firmly believes that Allah can hear even the quietest of prayers and thoughts.
Secondly, it is a sign of respect and manners in front of Allah. For, just as it is considered improper for the servant to raise his voice in front of his master, or the peasant in front of the king, even so it is improper that a slave raise his voice loudly in front of the Creator.
Thirdly, it is a means of achieving humility and humbleness, which is the essence of worship. The one who is humble does not ask except meekly, whereas the one who is arrogant asks loudly.
Fourthly, it is a means of achieving sincerity, since others will not notice him.
The companions of the Prophet understood the importance of saying a dua silently. Ibn Abbas stated: "A silent dua is seventy times better than a loud one!" And al-Hassan al-Basri said, "We used to be amongst a group of people (i.e., the Companions) who would never do any act in public if they could do so in private. And those Muslims would strive in making their dua, and not a sound would be heard from them! For they used to whisper to their Lord."
"Dua: The Weapon of the Believer"- Abu Ammar Yasir Qadhi, pp. 86-89