First of all, a Balloon is made of a highly elastic material like rubbers or nylons. All materials have a specific level of Elasticity or Plasticity (based on its nature except brittles where fracture occurs). To differentiate these properties, Hooke's law & thereby moduli of elasticity arrived. When you blow air inside the balloon, it expands thereby increasing the pressure (more specifically stress) within the balloon. When this pressure overcomes the ultimate tensile strength (breaking stress) of the balloon, it breaks. For now, there are two rules:
1) When you pop the balloon with a needle, the air inside tries to rush out (as it's in a high pressure in there). But, only a small area is provided by your needle (which is not enough for the air to rush out). The balloon begins to crack, thereby expanding the hole into a large fissure.
Now, Regarding the sound theme: The pressure inside (depends upon how much you blow) and outside (1 atm) the balloon creates a pressure difference in that area. This difference creates a longitudinal pressure wave (I'd say that as a small shock-wave) along with elastic energy of the balloon's material (while retaining to its original shape) which is perceived as sound.
2) When you open the mouth of the balloon, it has a lot of area for the air to get out. This doesn't pop the balloon. But indeed, the rushing damages the elasticity of balloon (atleast by some %).
But, you can reduce the sound by somewhat. After or before blowing the balloon, catch some of the rubber with your fingers. Then, pierce a hole in that small area and leave the air slowly. This doesn't pop it because, all you require is to prevent the hole from expanding into a fissure.
NB: In any of these ways, you can't prevent the sound emerging from inside. It always produces the "Psssssss....". Sound is produced until there's always a medium for its propagation.
Response to comment: Perhaps, you didn't notice what I've explained. What do you mean by "Both areas are equal". I haven't heard of such a needle equaling the area of mouth the mouth and a small point. Then, it won't have the name NEEDLE... Regarding different sounds - The tones you make is similar to whistles and other blowing instruments which is based on the simple air vibration. It could be done with a simple plastic chocolate wrapper which I've done many times during my high-school days :-)