Child asthma vs. Adult asthma or do you get both?

Depends. Most children with asthma still have asthma as adults, although the severity of asthma can change over time.
Label confusion. Many of those who wheeze during a variety of illnesses as infants and young kids grow out of this by school age (about 60%).They may be labeled as having asthma but I don't call this asthma. There are kids with true asthma that will show by 1yo in males & 4 in most females.They will have the tendency to have adult asthma. However, asthma is quite variable in intensity so label confusion is common.