To simplify it, pure mathematics is just mathematics. Applied mathematics on the other hand is using pure mathematics for a practical cause eg using mathematics to predict when the next El nino rains might hit us.
The easiest way to think of it is that pure maths is maths done for its own sake, while applied maths is maths with a practical use. But in fact, it’s not that simple, because even the most abstract maths can have unexpected applications. For example, the branch of mathematics known as “number theory” was once considered one of the most “useless”, but now plays a vital part in computer encryption systems. Applied maths tries to model, predict and explain things in the real world.
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. Pure mathematics, by contrast, is mathematics for its own sake