Why does a burp make a sound instead of sounding like a normal breath?

When we talk, we pass a 'stream' of air from our lungs over our vocal chords in the larynx to produce speech, song or other dignified noises. When we burp, the air is a 'bubble' of air originating from the stomach and as this passes through the throat it sets up a resonance independent from the vocal chords. Because there is little energy in the bubble of air it produces a deeper sound than our normal voice.

My son, however, is perfectly able to burp speech through controlling the shape of his mouth just as if he were talking.

People who have lost the use of their vocal chords (generally through having them removed due to cancer of the larynx) can learn to burp words.

That answer is basically right, but it is not the vocal cords that produce the sound in a burp. The vocal cords are situated in the windpipe so they are no anywhere between the stomach and the mouth. It is a flap of muscle/tissue at the top of the oesophagus (gullet or food pipe) that vibrates. Since this is not designed for producing sound, it is very slack and produces a low frequency noise.