Data Mining is a technique used in business and data science both, while data science is an actual field of scientific study or discipline. Data mining's goal is to render data more usable for a specific business purpose. Data science, in contrast, aims to create data-driven products and outcomes-usually in business context.
Data mining deals mostly with structured data, as exploring hug amounts of raw, unprocessed data is within the bounds of data science. However, data mining is part of what a data scientists might do, and it's a skill that's part of the science.