Bronchitis or asthma: how do you know which one your young child is suffering from?

Bronchitis = asthma. Adults who smoked develop chronic bronchitis. In children acute bronchitis is usually coughing, chest tightness, shortness of breath, wheezing with a viral respiratory infection (vri). In this case the lungs are over reacting to the irritation of the vri and over reactive/hper-reactive airways is the hallmark of asthma. Children who repeatedly get "bronchitis" probably have a mild form of asthma.
Likely asthma. Bronchitis is not that common in children, so if you were to "choose" between the two, it is likely asthma. But your pediatrician may need to evaluate the child and the symptoms.