The create new theories and try to solve problems with those theories. These two types of math are very different but they often overlap. Theoretical mathematicians try to discover new mathematical principles. They also try to find relationships between principles that have already been discovered. These discoveries are important because they may produce scientific and engineering achievements.
Applied mathematicians use theories and techniques to solve practical problems.
These problems are in areas of natural science, social science, engineering, business, and government. For example, mathematicians study the effects of new drugs on disease. They analyze results and use numbers to predict how the drugs will impact the disease. Other mathematicians might study how product costs will effect businesses. Some mathematicians are called cryptanalysts.