Singing can be thought of as the athletic use of the vocal cords and there are many ways to use the larynx for singing. With the vocal cords roughly capable of vibrating between 60 to 1,100 times per second when singing, we hear a wide range of sound.
When we hear a song on the radio, we are actually hearing the voice and speech at the same time. That is, we are hearing words (speech) at various pitches and volumes (voice). While in typical conversation we produce words at a very limited range of pitches and volumes, in singing we use a wide range of pitches and volumes. If we elimintated the speech aspect by taking away the words, we would have, in a sense pure singing - pure vocal cord vibrations.
At times, when singing in a western, classical music style, particularly opera, quite often we hear an almost pure, clear musical tone. For example, particularly at high pitches, the voice may have only a musical quality and the speech aspect cannot be discerned. Think of the soprano singing an aria at the top of her range. This style of pure tone singing requires the most regular and precise positioning and closure of the vocal cords.
By contrast there are popular or folk musicians with a very soft quality to their voice. Think of the country singer or steamy, cabaret singer. These particular types of musicians typically hold the back of their vocal cords apart so that air leak is part of the style. The musical quality might be more difficult to discern at times with this style.
There may be a whole lot of air leak or even irregular vibrations in some genres. Tuvan throat music or heavy metal screaming may use these qualities to attract listeners. The false vocal cords often vibrate to produce sound in some of these styles.
In singing, the emphasis on how the vocal cords vibrate generally rises in importance, above the words alone, because in singing, we wish to do more than just communicate information. By intentionally varying sound production to produce various tones, various volumes, and different degrees of air leak we are adding emotion to the pure information content of speech.