Research design is an integral part of research methodology, used to tell your reader which type of research design you have followed to resolve the research questions. Research design can be exploratory, explanatory, descriptive and others. It depends on the nature of the research problem. On the other hand, research methodology is a broad concept, which is used to tell your reader about research philosophy, research strategy, research design, research approach, sampling, data collection technique, data analysis, and other relevant techniques. The key discrepancy between research design and research methodology is that research design is an integral part of research methodology, but research methodology is not all about research design. Based on the nature of your study, you can choose research design in line with research strategy, method, data collection. Generally, research methodology is widely used in academic research, such as post-graduate thesis, Mphil and Phd Thesis. On the other hand, several journal articles, short communication, and white paper just introduced the research design to their reader. For details, please read contemporary books, such as Bryman & Bell (2015), Saunders et al., (2009), and so on
Research design is a subset of research methodology. Because methodology is much more than the design. For example, methodology may involve design, analysis, inferring etc.