Question: I'm studying statistics, and in a lot of textbooks, the regression formulas always refers to the functions themselves as f(x) parameterized by a,b,c or something. And they are often written $f(x; a,b,c)$, why can't we just write $f(x, a, b, c)$?

I would like to know perhaps the history of how this came about, and why there is this seemingly dual method of expressing the same idea. Is there some esoteric mathematical notation that a noob like me hasn't encountered yet, or did mathematicians of old just like doing things in different ways?

An example: A function that maps age of tree to height:

$$f(age; growth\\_rate) = age \* growth\\_rate$$ vs $$f(age, growth\\_rate) = age \* growth\\_rate$$

Answer: 
While the answers already given are correct, I suspect that they may be bringing in terminology and notations you are unfamiliar with. If so, let me try to recast the answer more simply.

You are correct that $f(x; a, b, c)$ is the exact same thing as $f(x, a, b, c)$. But the difference is in how we are using $x$ vs how we are using $a, b, c$. With $f(x, a, b, c)$, all four values are equally important, equally of interest. With the former, we are considering $f$ as a function of $x$ only. $a, b, c$ are simply values that we use in defining that function. By making these values variables too, the mathematics we do applies to all the different functions of $x$ that are defined by the various possible values of $a, b, c$, instead of having to work each particular example we need separately.

For example, when talking of linear functions, we will write things such as $$y = mx + b$$ You could consider this as defining $y$ as a function of three variables, $x, m, b$, but what is important is the dependence of $y$ on $x$. The slope $m$ and intercept $b$ are just values expressing which line is represented by the relationship between $y$ and $x$. So if want to find the root for $y = 0$, that is $x = \frac{-b}{m}$, not $b = -mx$ or $m = \frac{-b}x$.

That is what the notation $f(x; a, b, c)$ means. It is just telling you that $x$ is the one whose variance is important to us. The others we are going to consider fixed (although not currently known) values.
In fact there is a formal distinction between parameter and argument, which captures our subjective view.

Let $P$ be the parameter space (set of growth rates), $X$ the domain (set of ages) and $Y$ the codomain (set of heights). A family of functions from $X$ to $Y$ parameterised by $P$ is formally a function $f: P \rightarrow Y^X$, where $Y^X$ is the set of functions from $X$ to $Y$.

In other words, it is a function taking growth rates and returning maps from ages to heights.

Of course, you are correct that this naturally and uniquely corresponds to a function $F: P \times X \rightarrow Y$, that is, a function taking a growth rate and an age and returning a height. This natural isomorphism is succintly denoted by

$$ \left( Y^X \right)^P \approx Y^{P \times X}$$