Newton was the one who really pulled everything together in a fundamental way. Barrow (his advisor) developed a lot of Calculus as did Fermat and Descartes before him and Barrow suspected the fundamental theorem of Calculus but it was Newton who proved it (to the standards of the day) and that was the key to confidently solving differential equations. While the problems of differentiation and integration are what we think of as Calculus that's not really what it's about, it's really about solving differential equations and that's what Newton advanced and then applied to problems of physics of the day.