Differential equations, as well as partial differential equations describe rates of change under a variety of conditions.
Many aspects of nature as well as many processes are described using differential equations. Thus a basic (or better more through) understanding of differential equations is highly recommended for the application of mathematics in a modeling context.
Partial differential equations were the topic of one of my modules in my final year of my mathematics BSc, while differential equations, as dynamical systems, received attention during my second year of mathematics.
One simple everyday example of differential equations is the relationship between distance, velocity and acceleration. But other examples are the relationship between species in an ecosystem with the Lotka-Volterra model being one of the simpler, yet effective, predetaor prey models. A set of coupled differential equations is known as a dynamical system and also finds its application in chemical kinetics, describing the production rates of different species.
Partial dfifferential equations in are often found when changes in systems have multiple dependencies, for example a spacial dependency besides a time dependency. The temperature distribution in a solid, say a copper or steel block heated from one side would be a simple real life example of a problem that can be described easily using partial differential equations.
A shared feature between many dynamical systems as well as partial differential equation descriptions of real life problems is, that they can only be solved numerically. Often an analytical solution is extremely difficult to obtain or even impossible to obtain, making numerical methods the approach of choice in modelling and research.