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Pure math is always correct w/o error. Pure math delves into the realm of theory and generality. For example, the equation 3x+5=72. This is asking you to solve a very specific equation where x has one value and is very restricted in R^1. A pure math problem would ask you to prove integration by parts for all R^n, and even then, only as an undergrad. Applied math, on the other hand cares more about the real world and modeling. You might be asked to generate a graph based off of data or solve a very specific differential equation. However, applied math can only improve with the advancement of pure math. I.E. pure mathematicians create the patterns, rules, and tools of the trade while applied mathematicians take them out of theory and into practice.

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