How important are the role of asymptotes in a hyperbola?

The hyperbola given by the equation

$$ x^2 - y^2 = 1$$

looks like this:

_{(image source: wolfram alpha)}

At this scale, the hyperbola is virtually indistinguishable from the union of its asymptotes. Since lines are easier to understand than hyperbolas, at this scale it's much easier to understand the hyperbola in terms of its asymptotes.

Here is a graph of

$$\frac{x^2}{4}-\frac{y^2}{9}=1$$

at a large scale.

I would also include the graphs of the asymptotes

$$ y=\pm \frac{3}{2}x$$

but I think the result would be obvious.

Yes, the asymptotes tell you the global behavior of the hyperbola, and in general it is extremely useful to be able to explain the global behavior of something complicated using simpler things that you already understand well (such as linear functions).

Specifically, they tell you that a hyperbola becomes linear far away from its center.