This is a slight elaboration on Jim Humphreys' answer. As far as I understand, at least one of the motivations of Hermann Weyl to develop the representations of compact Lie groups (such as $U(n)$, $SO(n)$) was to generalize the classical invariant theory and to apply this to differential geometry (tensor calculus) and closely related questions of general relativity. In these fields representation theory often allows to simplify complicated computations with tensors. For example if you know a priori that some complicated tensor you are interested in must be say $O(n)$-invariant, then if the space of invariant tensors of that type must vanish for representation theoretical reasons, then your complicated expression also vanishes.

To give a concrete example of application of that sort (which in fact is more involved than what I just described), let me remind the so called tube formula due to Weyl himself. Let $(M,g)$ be a Riemannian manifold isometrically imbedded into Euclidean space $\mathbb{R}^N$. Consider the volume in $\mathbb{R}^N$ of the $\varepsilon$-neighborhood of $M$ inside $\mathbb{R}^N$ as a function of $\varepsilon\geq 0$. Weyl proved:

1. This function is a polynomial in $\varepsilon$.

2. The coefficients of this polynomial, after appropriate normalization, are intrinsic invariants of the Riemannian structure $g$, i.e. are independent of the isometric inmedding into $\mathbb{R}^N$. More precisely they are polynomials in the components of the Riemann curvature tensor of $M$.

The proof of (1) is both elementary and easy. Part 2) is indeed non-trivial.Weyl's proof used the invariant theory of $O(n)$.

This is a slight elaboration on Jim Humphreys' answer. As far as I understand, at least one of the motivations of Hermann Weyl to develop the representations of compact Lie groups (such as $U(n)$, $SO(n)$) was to generalize the classical invariant theory and to apply this to differential geometry (tensor calculus) and closely related questions of general relativity. In these fields representation theory often allows to simplify complicated computations with tensors. For example if you know a priori that some complicated tensor you are interested in must be say $O(n)$-invariant, then if the space of invariant tensors of that type must vanish for representation theoretical reasons, then your complicated expression also vanishes.

To give a concrete example of application of that sort (which in fact is more involved than what I just described), let me remind the so called tube formula due to Weyl himself. Let $(M,g)$ be a Riemannian manifold isometrically imbedded into Euclidean space $\mathbb{R}^N$. Consider the volume in $\mathbb{R}^N$ of the $\varepsilon$-neighborhood of $M$ inside $\mathbb{R}^N$ as a function of $\varepsilon\geq 0$. Weyl proved:

1. This function is a polynomial in $\varepsilon$ for $\varepsilon \geq 0$ small enough.

2. The coefficients of this polynomial, after appropriate normalization, are intrinsic invariants of the Riemannian structure $g$, i.e. are independent of the isometric inmedding into $\mathbb{R}^N$. More precisely they are polynomials in the components of the Riemann curvature tensor of $M$.

The proof of (1) is both elementary and easy. Part (2) is indeed non-trivial. Weyl's proof used the invariant theory of $O(n)$.