These words mean "following one another in order," "marked by logical sequence." Consecutive suggests uninterrupted succession, whereas successive means "following in a regular sequence" with one thing in relation to another. Four consecutive months means four months in a row without interruption: March, April, May, and June. Four successive months would correspond to calendar order but might mean March, May, July, and October. See also continual.
Dictionary of problem words and expressions. Harry Shaw. 1975.