The line between pure and applied math isn’t squeaky clean, and their respective contents vary a bit from school to school. Generally speaking, pure math encompasses more math subjects, including the more esoteric ones, while applied math focuses on more practical techniques in fewer number of fields where they can be applied. Even on the same subject, a pure math major will learn more about the theoretical aspects of the subject, while an applied math major will focus more on the applications and less on abstractions.
For classical cryptography specifically, the most relevant math for now is number theory and probability. Number theory is obviously a pure math subject and probability is a subject in both pure math and applied math. However, cryptography is about to experience seismic and foundational changes. Quantum computing will render the current technique that relies on prime number factorization (a number theory subject) obsolete.