As a native English speaker, I see little difference between "firmly" and "strongly". That said, I tend to hear "strongly" used when something is strong: a machine, a person, materials. "Firmly" would not fit well in that circumstance. "Firmly" is more often used when a parent is being "firm" in disciplining or explaining rules to their child. (The relationship could also be teacher-student, boss-worker, or other relationships with a difference in power.) For more examples: http://dictionary.cambridge.org/dictionary/british/firmly http://dictionary.cambridge.org/dictionary/british/strongly