In the name of Allah the most compassionate, the most merciful.

Quran is the blessing and a mercy of Allah upon mankind. Man can benefit from holy Quran in a number of ways. 
Many Muslims have weak or no command at all on the Arabic language yet they endeavor to read recite and listen to Quran. 

Listening to Quran has numerous benefits as well but one needs to listen attentively.

Listening to Quran intently is a very beneficial and recommended activity for Muslims even if one cannot understand it.

Allah says in the Qur’an:
And when the Qur’an is recited, give ear to it and pay heed, that you may obtain mercy. [7:204]
Ibn Taymiyah said: “By means of this listening, Allah guides His slaves and sets straight their affairs in this world and the Hereafter.”
Al-Sadi said,
As for listening to it, this means that one hears it with an attentive heart, pondering the meaning of what one is listening to. The one who adheres to these two commands when the Book of Allah is recited will attain a great deal of good, beneficial knowledge, ongoing and renewed faith, increased guidance, and insight into his religion. [Tafsir Al Sadi]

Hence it is an indication that a Muslim must listen to Quran very intently and attentively, without any distractions. It is not something you play on background. One has to give it proper respect.
This leads to being enveloped by the mercy of Allah, and other benefits.
Allah mentions how a group of Jinn once listened intently to the Qur’an being recited by Prophet Muhammad (peace be upon him).

After they had heard it, they returned to their people, telling them about it, and warning them to heed it:
And [mention, O Muhammad], when We directed to you a few of the jinn, listening to the Qur’an. And when they attended it, they said, “Listen quietly.” And when it was concluded, they went back to their people as Warners. [46:29]

In the chapter of the Qur’an titled “Al-Jinn”, Allah says:
Say, (O Muhammad), “It has been revealed to me that a group of the Jinn listened and said, ‘Indeed, we have heard an amazing Qur’an’.” [72:1]

The Superiority of Reciting Qur’an Over Listening
It is indeed without any doubt that listening to Quran has its benefits and is rewarding but the recitation holds its superiority over it. Reciting a Quran with the tongue has more benefits.
This is because more of the human senses are engaged when a Muslim recites the Qur’an aloud than when they just listen to it.
 It includes tongue, ears, as well as eyes. 
The heart, as well as the body, connect more deeply to the Qur’an when a Muslim recites it aloud, with full focus and attention than when they passively listen to it.

The Prophet also said:
If anyone recites a letter from Allah’s Book, he will be credited with a good deed, and a good deed gets a tenfold reward. I do not say that Alif-Laam-Meem is one letter, but Alif is a letter, Laam is a letter and Meem is a letter. [Jami Al Tirmidhi]
Therefore if you have an opportunity to choose from reciting and listening always choose to recite. And if the circumstances don't permit it then avail the mercy of Allah by listening to it attentively.
We can conclude that it is a good deed to remain attentive and silent while listening to Quran recitation. It humbles the heart and increases the Iman of a person.

A Muslim should do both good deeds: reciting the Qur’an with their tongue, as well as listening to it quietly, with full attention.

Both these honorable actions reinforce each other.

The Prophet said:
The one who recites the Qur’an while he is proficient in its recitation will be with the honorable and obedient scribes (angels), and the one who recites the Qur’an while it is difficult for him to recite, will have a double reward. [Sahih Al Bukhari & Muslim]

May Allah [swt] rewards us.