What is the possible advantage that immunotherapy might have?

Many. By changing your immune system to make you less allergic, rather than just treating the symptoms like medications do, immunotherapy is far more effective than medications. You'll also be able to tolerate unexpected exposures (going to the home of a friend of a friend who has cats) without pre-medicating. It reduces the risk of developing or worsening asthma, whereas medications alone do not.
Immunotherapy perks. Immunotherapy usually refers to the injection of extracts of environmental allergens, such as pollens and dust, to reduce sensitivity to these allergy triggers. As opposed to medications, which reduce symptoms but do not suppress the underlying allergies, immunotherapy modifies the person's allergic response, allowing them to tolerate increased exposure with fewer symptoms and less drug use.