Biomedical and biochemical research both investigate the chemistry of life. While they overlap in many areas, they are not one and the same. Biomedical research is broader in terms of the techniques and disciplines it employs, whereas biochemical research is broader in terms of its aims and objectives. Moreover, while biochemistry is fundamental to biomedicine, not all biochemical research is biomedical in nature.
Biochemists are scientists who seek to understand the chemical processes of life. They want to know what kinds of chemical reactions take place in living organisms, how those reactions and processes are regulated and controlled, and how these processes affect an organism's observable characteristics. They employ a wide variety of tools in their research, and in many areas biochemistry overlaps with the closely related field of molecular biology. Biochemical research may be basic (aimed at understanding nature) or applied (aimed at using scientists' understanding of nature to design a useful product like a drug).
Biomedical research seeks to solve medical problems. Scientists in this field want to cure diseases like cancer or find out what causes them and prevent them from occurring in the first place. There are many different ailments for which medicine does not yet have a cure, so biomedical research is very broad. It is also cross-disciplinary, because developing new medical technologies requires a diverse array of approaches. Molecular biologists, for example, may work on finding new drug targets while chemists design drugs to hit these targets.
Like biochemical research, biomedicine may also be basic (aimed at understanding a disease) or applied (aimed at finding a way to cure it). Both basic and applied sciences are important because you need a basic-science understanding of a disease as a foundation for applied science research. Although biomedical research makes use of biochemical techniques, it does not rely exclusively on methods from that field. Synthetic organic chemists, medicinal chemists, doctors and molecular biologists (among others) also play key roles in biomedical research. In terms of the techniques employed, biomedical research is broader than biochemistry.
While biochemistry is more limited in terms of the techniques it uses and the processes it seeks to understand, it is broader in terms of the kinds of organisms researchers study. For example, some biochemists study the biology of marine organisms or work on turning algae into biofuel. Neither of these fields of study falls within the confines of biomedicine. Ultimately, biomedicine and biochemistry are both concerned with the chemistry of life. Biomedicine, however, seeks to solve human medical problems using tools from many fields, while biochemistry answers questions about chemical processes both in humans and other organisms.