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Sorry if this is too much of a soft question or if it is too broad for this site.
As a person who has much more experience with math than biology, I've always been really interested in the mathematical side of biology. As I study math in biology, however, I am left with the question of "how is this actually useful?"
What kind of unique insight can mathematical models give us about biological systems that we couldn't just find experimentally?
This question is broad.
There are plenty of uses of mathematical modelling in biology.
Wikipedia
You may want to read the wikipedia article: Mathematical and Theoretical Biology.
Books
There are tons of books on theoretical biology. I am personally working in the field of population genetics, a field where much of the theory has been expressed mathematically. Here are book recommendations for this particular field.
On Biology.SE
We have a tag for mathematical-models (you used it in your post). You may want to see what kind of posts are tagged for mathematical-models.
To repeat @fileunderwater's comment, What physics knowledge can be applied to biology of organisms and ecosystems and When are population dynamics models useful? are examples of posts you may want to read.
What prevents predator overpopulation? will give you an example of an easy and classical mathematical model to describe population dynamics of predators and preys.
Personal historical/philosophical note: At some point physicists have started to describe the laws of nature with mathematical expressions and at that time many physicists didn't quite like the idea that physics could be described by math. Biology has gone through the same phase but much later. Early mathematical biologists developed the modern science of statistics (Francis Galton, Karl Pearson, Ronald Fisher). Biologists used existing mathematical tools such as diffusion processes (Kimura). Note that part of the work in the field of diffusion equations have been done by physicist (Einstein solved the 2D equations for diffusion processes). Note also that the dynamic of diffusion processes called brownian motion have been named after a biologist. Today, biology produces massive amount of data that require good computational skills to treat. The time when biologists had only to know how to recognize species is over, now there are few biologists left that aren't able to code in at least one programming language.
Models are often used to calculate the probability of certain biological outcomes. Such predictions are great help for prioritization of further studies or making other decisions.
Example: EPA uses an array of in vitro assays for fast and cheap analysis of thousands of potentially toxic environmental compounds (ToxCast and Tox21 projects). Next, they use the obtained data to build mathematical models to prioritize chemicals for further more expensive and slow animal studies or to decide on administrative actions to restrict usage of high priority (most dangerous) chemicals. Their prioritization models integrate toxicity data and complex exposure models (including manufacturing and usage logistics, possibility of accidental exposure etc).
