I noticed that both terms referred to the study of mathematics. Is there a difference in usage between the two terms?
数学 is (pure) mathematics.
数理 refers to mathematical sciences, "group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper." In other words, 数理系の研究者 use difficult mathematics to solve actual problems that matter in human life, such as cryptography or machine learning.
I'll illustrate using examples.
The Kyoto University Research Institute of Mathematical Sciences (KURIMS, 京都大学数理解析研究所) is an example of a 数理研究所, and you see their research span many areas. They have pure math research (like algebra and analysis), but they also work in mathematical physics, fluid dynamics, algorithms, and theoretical computer sciences. Similarly, 名古屋大学多元数理研究科 conducts research in pure mathematics, applied mathematics, and "mathematical sciences" (i.e., computational fluid dynamics, mathematical physics, theoretical computer science, etc).
OTOH, the Hokkaido University Department of Mathematics is "just" a math department, and their research is mainly pure mathematics. They do have a 数理科学 section, but all applied mathematics/"mathematics sciences" are confined to this section only.
So, the conclusion is: the practical difference is not that great. 数学 is a true subset of 数理科学. 数学 is more or less the pure math part, but 数理科学 includes what you'd call "computational methods" and "applied mathematics": specifically, this usually includes the fields of operations research, computational engineering methods, mathematical physics, and theoretical computer science.
Practically, though, most Japanese universities have departments of 数理科学 (at least at the graduate level), not (just) departments of 数学. The difference is not notable. Even if you work in mathematical physics or TCS, non-professionals would not care if you pass around as a mathematician (数学者) instead of a mathematical scientist (数理科学者) - they likely don't understand your work anyways. If you're communicating with professionals, you'd likely be telling them your specific sub-field as well (e.g. 計算機科学 "theoretical computer science", 数理物理学 "mathematical physics", 複雑系科学 "complex systems"). Whether they will consider you colleagues in mathematics is a totally different question :-)