Asymptotic analysis tends to be useful when you expect the function to take large values as input, but don't necessarily know how large they will be. For instance, knowing that the asymptotic complexity of sorting is $O\left(n\log n\right)$ isn't all that useful when you are solving a list of 7 elements; it's mainly useful when considering sorting lists of 1000000000 elements (where you can infer that e.g. $O\left(n^2\right)$ would be too expensive but $O\left(n\log n\right)$ is not).

I guess it is also sometimes useful as a sort of "interpolation strategy". Like if you want to know the properties of a function, you can first asymptotically characterize its limiting properties, and that covers a good chunk of the information about the function, leaving you only with a bounded area of uncertainty. Then you can analyze how it shifts between its asymptotes in the middle.