Biochemistry and chemistry both share a great deal in common; biochemistry, however, is usually considered a subfield of biology rather than of chemistry, albeit one that requires a great deal of chemistry knowledge. The difference between biochemistry and chemistry boils down to the emphasis in each, together with the different systems they study and the approaches they use when investigating these systems.
Biochemistry studies the chemistry of life -- how life works at the molecular level, what kinds of chemical reactions occur in cells, and how these reactions give an organism its observed characteristics. The field overlaps in many areas with disciplines like molecular biology and genetics. A great deal of chemistry knowledge is involved -- especially knowledge of organic chemistry, since most of the important reactions that take place in organisms involve carbon compounds. Inorganic chemistry is involved as well, however, since many enzymes have metal-ion cofactors.
Chemistry comprises several distinct subdisciplines, namely, physical, inorganic and organic chemistry. In some ways it's a broader field than chemistry, since it's concerned with the structure and behavior of matter and the types of reactions compounds can undergo. The dividing line between biochemistry and chemistry is a little fuzzy, but in general, chemists are interested in designing useful new materials, finding more efficient ways to synthesize existing materials, or understanding why substances have the property that they do, while biochemists use chemistry to understand why and how certain processes take place in living organisms.
Historically, chemistry was by far the more important field in industry. Starting with the invention of recombinant DNA technology in the early 1980s, however, a new industry called biotechnology was born, and today there are an expanding variety of opportunities for biochemists in academic research, pharmaceuticals, biofuels and biotechnology. In biomedical research, chemists generally work to design new drug molecules, while biochemists try to understand the processes taking place in an organism and how they might furnish possible drug targets the chemists can exploit. They also design assays to test drug compounds for activity.
Course of Study
If you are choosing a major in college and are thinking of picking either biochemistry or chemistry, here are some factors you might consider. You'll need to learn a lot of organic chemistry to succeed in either discipline; in a chemistry major, however, you'll spend a lot of time on physical and inorganic chemistry, whereas in biochemistry you'll devote more of your attention to molecular biology. Basically, the emphasis in each major is somewhat different.
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