Quantitative Finance integrates Mathematics, Statistics, and Computer Science with the objective of practical application towards financial markets. One of the primary concerns of quantitative finance is risk. Some of the primary concerns of Quantitative Finance include:

The pricing of securities, especially derivative securities

Risk
A large amount of work in quantitative finance relies on the following two assumptions:

Markets are efficient

There is no free lunch (no arbitrage assumption)
Two common approaches to the pricing of derivatives are the riskneutral pricing framework and the differential equations approach. Binomial Trees are an example of riskneutral pricing, and the BlackScholes equation is an application of partial differential equations.
The differential equations approach often leads to closedform solutions, leading to straightforward pricing formulas such as the BlackScholes formula. These are referred to as analytical solutions. In other cases, differential equations have no known analytical solution, and thus must be solved numerically. For instance, there is no closedform solution for the price of an American Put option; though solving for the price of an American Put numerically is not computationally intense.
The riskneutral approach can also entails use of stochastic calculus. While analytical solutions are possible, numerical solutions are more common in the riskneutral approach. The riskneutral approach is more flexible than the differential equations approach and can be used to price derivitives that cannot be valued using differential equations or BlackScholes.
Stochastic calculus often requires a "change of measure" between the "real world" of probabilities and the "riskneutral world" of probabilities.