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Why use mod-arg form instead of real-imaginary?

The two uses i can think of is when multiplying complex numbers and also it provides us with a geometric perspective?

Does anyone know any other specific areas where it's useful?

Don't worry about difficulty of such areas.

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    In the elementary theory of Fourier series it becomes necessary to have formulas for $\sum_{n=0}^N \cos (nx)$ and for some related creatures. It's easy when you write $ \cos nx=(e^{inx}+e^{-inx})/2$ as you now just have to sum a pair of finite geometric series. – 2017-01-16

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Presumably you mean why would one use $z=r(\cos\theta+i\sin\theta)$ instead of $z=x+iy$ in a problem?

If so, one easy example would be in calculating powers of complex numbers. That is, essentially solving equations of the form $z^n=w$.

Try finding $z\in\mathbb{C}$ such that $z^4=-1+i\sqrt3$ with both forms and see for yourself which one is easier and faster.