How could I know if my trouble breathing is asthma or just allergies?

See a doctor. Spirometry can demonstrate obstructive lung disease that is reversable. This will rule in or out asthma for you.
Albuterol response. There are lots of ways to distinguish asthma from allergic rhinitis. Asthma symptoms get better with albuterol; nasal symptoms don't. Spirometry & measurement of breath Nitric Oxide (feno) are useful tests for asthma. They help make the diagnosis as well as determine severity. These tests are easily performed in the office of a pulmonologist or an allergist.
Brathing asthma. Allergies may cause breathing problems from the nose. Asthma will make it difficult to breathe from the lungs. Asthma is more of a debilitating medical problem... It limits activities, etc... Allergies are a nuisance..