Consistently is a adverb meaning "steadfastly," "unwaveringly," "without change": "He is consistently behind in his payments." "This student consistently objects to all authority." Constantly means "unceasingly," "without interruption," "perpetually": "My concentration was constantly broken by the loud ticking of a clock." "My mail is constantly filled with second-class matter." Related words that might express meaning more exactly than either consistently or constantly are perseveringly, persistently, enduringly, continually, and unceasingly.
Dictionary of problem words and expressions. Harry Shaw. 1975.