What is the importance of $\sinh(x)$?

The other answers are spot-on, but I'll take the opportunity to point out yet another real-world application of hyperbolic trig functions.

A catenary is the curve you get by hanging a chain of uniform density by its two endpoints:

$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $

Catenaries will arise, for example, in suspension bridges and power cables hung from transmission towers. These curves and their properties are nicely described using the hyperbolic trig functions. As an interesting aside, square wheels will "roll" smoothly over a surface of inverted catenaries$^\dagger$, provided that the arclength of each catenary is equal to the side length of the square.

$\text{ }$

This is all to say: one can imagine that engineers and architects would find interest in studying these curves in certain situations. See this paper as an example, where one repeatedly encounters the hyperbolic sine and cosine functions.

$^\dagger$The intrigued reader can see this in action here. The particularly intrigued reader might be asking whether a suitable road exists for other wheel shapes, and this question is addressed extensively by Hall & Wagon in their 1992 paper Roads and Wheels, again making heavy use of these hyperbolic functions.

The hyperbolic sine (and the hyperbolic cosine) arises naturally in the solution to the differential equation

$$f''=f$$

which is

$$f(x)=a\sinh(x)+b\cosh(x)$$

This point of view uses the linear algebra:

The set of functions $f:\Bbb R\to \Bbb R$ is a linear space usually denoted $\mathscr F(\Bbb R,\Bbb R)$ or simply $\mathscr F(\Bbb R)$ and it's a direct sum of the subspace of even functions $\mathscr E(\Bbb R)$ and the subspace of odd functions $\mathscr O(\Bbb R)$:

$$\mathscr F(\Bbb R)= \mathscr E(\Bbb R)\oplus \mathscr O(\Bbb R)$$ which means that any function $f$ is a sum of an even function and an odd function and this writing is unique

$$f(x)=\underbrace{\frac{f(x)+f(-x)}{2}}_{\text{even part}}+\underbrace{\frac{f(x)-f(-x)}{2}}_{\text{odd part}}$$

Now for the $\sinh$ function we can see easily that it's the odd part of the exponential function.