What does it mean to have a cough asthma? how is that different from asthma?

Cough variant asthma: Cough variant asthma is commonly a mild form of asthma. The hallmark of traditional asthma is a whistling noise with breathing known as a wheeze. In cough variant, there is less frequently a wheeze and more commonly a cough. The common medicines for asthma usually treat it.

May not be different: Asthma can cause a cough, and coughing can cause airway constriction ...

Less intense: Cough variant asthma was not actually described until well into the late 80's or so. It often shows up as a cough resistant to cough meds but helped by asthma meds. Linkage to families where other members have true asthma is common. In my family 2 sibs had bad asthma, mom did not wheeze until >70 & my cough variant asthma was finally evident when they had a name for it.