Rhum isn’t simply another way to spell rum. That additional “h” means something.
Rum (sans an “h”) is the overarching term for all styles of rum, including dark and full-bodied Guyanese demerara rums like El Dorado, Jamaican rums like Appleton Estate, deep and rich Guatemalan rums like Zacapa, crisp Cuban rums like Havana Club, and Venezuelan rums like Diplomatico.
Rhum, on the other hand, is a much more specific term. An abbreviation from the term rhum agricole, this type of rum can only come from Martinique. And while most rums are distilled from fermented molasses, rhum is made from fresh pressed sugar cane juice. On the island, sugar cane is grown specifically for the production of rhum agricole, and the fields are managed and controlled like vineyards.