What are the differences between child asthma vs. Adult asthma?

Young or old adult? Young adults - 30s & 40s - with asthma have had it since childhood or had it during childhood, then had resolution for a decade or so, followed by recurrence. They might recall only having bronchitis often when younger. This is a continuation of a chronic, usually allergy-based illness from childhood. Asthma starting after age 50 is a different disease & usually not allergic in origin.
Child more allergic. Children with asthma are more likely to have a significant allergic component to their illness. However, allergy can be a factor at any age. Both childfren and adults respond well to medications, and the agents used are similar. Typically attacks in children evlove more quickly than those in adults, and children usually respond faster to treatment.
Age... The disease process is the same for children and adults and that is recurrent episodes of bronchospasm and airway inflammation. It used to be thought that kids had more of an allergic etiology and adults had more exacerbations with infections but there is so much overlap that this is really not a useful distinction.