By definition, applied mathematicians focus on the real-world uses of mathematics. Engineering, economics, physics, finance, biology, astronomy—all these fields need quantitative techniques to answer questions and solve problems. However, pure mathematics is mathematics for its own sake. Nonetheless, there are some overlap between them.
Mathematics in one!
The division into pure and applied mathematics is artificial and mostly helps people to easier get grant money if he/she claims is an applied mathematician.
Traditionally, main branches of applied mathematics include: Differential Equations, Numerical Analysis, Optimization (linear and nonlinear), and Statistics. One can also add Harmonic Analysis (wavelet theory developed in the past 35 years), Dynamical Systems (fractals, developed in the past 40 years), as well as compressed sensing, and the list goes on and on. Each of these disciplines has profound theoretical (pure mathematical) foundations.
Recent developments in computation theory and IT revolution show, on daily basis, that traditional pure mathematical disciplines, as abstract algebra, geometric algebra and number theory, harnessed to coding theory, are nothing less than applied mathematics.
BTW, the first line of my statement can be attributed to a prominent Fields Medalist (1966), Stephen Smale who very successfully worked in both areas of pure and applied math.
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