Science has long been viewed as a key driver for economic growth and rising standards of living. Knowledge about how scientific advances support marketplace inventions is therefore essential for understanding the role of science in propelling real-world applications and technological progress. The increasing availability of large-scale datasets tracing scientific publications and patented inventions and the complex interactions among them offers us new opportunities to explore the evolving dual frontiers of science and technology at an unprecedented level of scale and detail. Yet we lack suitable visual analytics approaches to effectively analyze such complex interactions. Here we introduce InnovationInsights, an interactive visual analysis system for researchers, research institutions, and policymakers to explore the complex linkages between science and technology, and identify critical innovations, inventors, and potential partners. The system first identifies important associations between scientific papers and patented inventions through a set of statistical measures introduced by our experts from the field of Science of Science. A series of visualization views are then used to present these associations in the data context. In particular, we introduce the Interplay Graph to visualize patterns and insights derived from the data, helping users effectively navigate citation relationships between papers and patents. It thereby helps them identify the origins of technical inventions and the impact of scientific research. We evaluate the system through two case studies with experts followed by expert interviews. We further engage a premier research institution to test-run the system, helping its institution leaders to extract new insights for innovation. Through both case studies and the engagement project, we find that our system not only fulfills our original purposes of design, allowing users to better identify the sources of technical inventions and understand the broad impact of scientific research; it also enables an array of new applications for researchers and research institutions, ranging from identifying untapped innovation potentials within an institution to forging new collaboration opportunities between science and industry.